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Internal Direct Products of Groupoids
Journal of Algebra 217, 599᎐627 Ž1999. Article ID jabr.1998.7812, available online at http:rrwww.idealibrary.com on
Internal Direct Products of Groupoids K. E. Pledger Mathematics Department, Victoria Uni¨ ersity of Wellington, P.O. Box 600, Wellington, New Zealand Communicated by Leonard Lipshitz Received June 30, 1998
When generalizing the internal direct product from groups to all groupoids Žbinary systems., it has been customary to imitate the group case by making restrictions, especially the existence of an identity element. This article develops what seems a natural basic definition of internal direct product, then uses it as a background against which to compare more popular restricted versions. 䊚 1999 Academic Press
1. PROLOGUE The fruitfulness of Jonsson and Tarski’s work w5x on direct product ´ decompositions is very clear from Willard’s summary w7, pp. 130᎐131x. Yet w5x includes a curious oversight concerning the definition of internal direct products of groupoids. That definition generalizes the familiar group case, which can be described as follows without mentioning normal subgroups. A group G may be the internal direct product of subgroups G1 and G 2 . In that case each element of G is uniquely expressible as x 1 x 2 with x 1 g G1 , x 2 g G 2 ; and the product of any such x 1 x 2 and y 1 y 2 can be written Ž x 1 y 1 .Ž x 2 y 2 .. The connection with the ordinary external direct product G1 = G 2 is that the mapping : Ž x 1 , x 2 . ¬ x 1 x 2 is an isomorphism from G1 = G 2 onto G. If G is an external direct product H1 = H2 of groups, then it is also an internal direct product as follows. Let e1 be the identity element of H1 , and e2 of H2 , then let G1 s H1 = e2 4 and G 2 s e14 = H2 . It is easy to 599 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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K. E. PLEDGER
check that G is the internal direct product of G1 and G 2 , although the isomorphism is the notationally clumsy
Ž Ž x 1 , e 2 . , Ž e1 , x 2 . . ¬ Ž x 1 , e 2 . Ž e1 , x 2 . s Ž x 1 , x 2 . . Such arguments depend so vitally upon e1 and e2 that past generalizations of the internal direct product from groups to groupoids have treated identity elements as indispensable. The classic statement of that belief occurs within the book w5x of Jonsson and Tarski. Their algebras are ´ groupoids with multiple operators, but the essential problem with internal direct products lies in the groupoid structure. On p. 7 they make the following claim. ‘‘On the other hand, the definition of a winternal x direct product implicitly involves the notion of a zero element, and it cannot be applied to systems without a zero unless we agree to use the term ‘‘subalgebra’’ in a more general sense. What is even more important, a detailed examination of the proof of 1.5 wcharacterizing internal direct productsx reveals that condition 1.1Žii. wexistence and properties of zerox plays an essential role in this proof. It can easily be shown by means of examples that, in general, Theorem 1.5 does not apply to systems A in which either part Žii⬘. or part Žii⬙ . of this condition fails; for it may then happen that A is isomorphic to the cardinal wi.e., external directx product of two systems B and C, without containing any subsystem isomorphic to B or C. We can thus say that only by restricting ourselves to systems which satisfy 1.1Žii. are we in a position to introduce an adequate notion of an inner direct product.’’ Example 7.1 illustrates the difficulty they described and firmly blamed on the lack of an identity element. The strong but unproved statement of Jonsson and Tarski seems to have started a tradition, and not only in later ´ papers involving the same authors w2᎐4x. Łos´ w6x uses more general algebras, but imposes conditions from which he can derive essentially the existence of the groupoid identity element as a theorem. That tradition is renounced by Definition 2.2 below, whose consequences do not include the existence of an identity element. It seems the most natural and economical definition of internal direct product. ŽEvidently Cohn thinks so, as his w1, p. 56, Exercise 11x gives the same definition; but he then reverts to tradition by citing w5x..
2. FUNDAMENTALS The notation is reversed Polish. Various groupoids use the same operator symbol  , so the result of the appropriate binary operation on x and y is written xy  . Also the notation introduced in each definition is assumed from then on.
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DEFINITION 2.1. The external direct product of groupoids Ž G1 ,  . and Ž G 2 ,  . is Ž G1 = G 2 ,  ., where ᭙ x 1 , y 1 g G1 ,
᭙ x 2 , y 2 g G2 ,
Ž x1 , x 2 . Ž y1 , y 2 .  s Ž x1 y1  , x 2 y 2  . . DEFINITION 2.2. A groupoid Ž G,  . is the internal direct product of its non-empty subgroupoids G1 and G 2 if and only if the mapping : Ž x 1 , x 2 . ¬ x 1 x 2  is an isomorphism from G1 = G 2 onto G. Prescribing that G / ⭋ is a minor convenience. Of course is just the mapping which interprets the binary operator symbol  in G. Next, the isomorphism y1 can be used to define the projections ␣ 1 and ␣ 2 from G onto G1 and G 2 . DEFINITION 2.3. With the notation of Definition 2.2, ᭙ x g G, x y1 s Ž x ␣ 1 , x ␣ 2 . . From Definitions 2.2 and 2.3 easily follow G␣ 1 s G1 and G␣ 2 s G 2 . Hence an element of G1 may be written either x 1 or x ␣ 1 as convenient, and a similar remark applies to G 2 . This is assumed in, for example, the proof of Theorem 2.5Žb.. Also the consequence that G1 ␣ 1 : G1 and G 2 ␣ 2 : G 2 is useful from Theorem 3.2 onward. THEOREM 2.4. ᭙ x 1 , y 1 g G 1 , ᭙ x 2 , y 2 g G 2 , x 1 x 2  y 1 y 2  s x 1 y 1  x 2 y 2  ; i.e., elements can be combined componentwise. Proof. x 1 x 2  y 1 y 2  s Ž x 1 , x 2 . Ž y 1 , y 2 .  s Ž x 1 , x 2 . Ž y 1 , y 2 . 
2.2
s Ž x1 y1  , x 2 y 2  .
2.1
s x 1 y 1  x 2 y 2  .
2.2
The second paragraph of Section 1 has now been formalized for general groupoids. Such pedestrian results are easily obtained by going back and forth to the external direct product, but for faster progress an internal direct product can be characterized purely by intrinsic properties. THEOREM 2.5. Suppose a set G is closed to a binary operation  and unary operations ␣ 1 and ␣ 2 . The groupoid Ž G,  . is the internal direct product of G␣ 1 and G␣ 2 if and only if the algebra Ž G,  , ␣ 1 , ␣ 2 . has the
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following properties: Ža. ᭙ x g G, x ␣ 1 x ␣ 2  s x Žb. ᭙ x, y g G, x ␣ 1 y␣ 2 ␣ 1 s x ␣ 1 and x ␣ 1 y␣ 2 ␣ 2 s y␣ 2 Žc. ᭙ x, y g G, x ␣ 1 y␣ 1  s xy ␣ 1 and x ␣ 2 y␣ 2  s xy ␣ 2 . Proof. Write G␣ 1 s G1 and G␣ 2 s G 2 . First, assume Ž G,  . is the internal direct product of G1 and G 2 . Then:
Ž a.
x s x y1 s Ž x ␣ 1 , x ␣ 2 .
᭙ xgG,
2.3
s x ␣1 x ␣2 
Ž b.
᭙ x 1 g G1 ,
2.2
᭙ y 2 g G2 ,
Ž x 1 , y 2 . s Ž x 1 , y 2 . y1 s x 1 y 2 y1 s Ž x 1 y 2 ␣ 1 , x 1 y 2 ␣ 2 .
2.2 2.3
so x 1 s x 1 y 2 ␣ 1 and y 2 s x 1 y 2 ␣ 2
Ž c.
᭙ x, ygG,
Ž xy ␣ 1 , xy ␣ 2 . s xy y1 y1
s x
2.3 y1
⬗ y1 is an isomorphism by 2.2
y 
s Ž x ␣ 1 , x ␣ 2 . Ž y␣ 1 , y␣ 2 . 
2.3
s Ž x ␣ 1 y␣ 1  , x ␣ 2 y␣ 2  .
2.1
so xy ␣ 1 s x ␣ 1 y␣ 1  and xy ␣ 2 s x ␣ 2 y␣ 2  . Conversely, assume Ža. ᎐ Žc. are satisfied. Define : G ª G1 = G 2 by x s Ž x ␣ 1 , x ␣ 2 .. Then x s y
«
x ␣ 1 s y␣ 1
«
x ␣ 1 x ␣ 2  s y␣ 1 y␣ 2 
«
xsy
and
x ␣ 2 s y␣ 2
by Ž a . . ⬖ is injective.
Also ᭙ x 1gG1 ,
᭙ y 2 g G2 , s Ž x1 , y2 .
x 1 y 2  s Ž x 1 y 2 ␣ 1 , x 1 y 2 ␣ 2 . by Ž b . . ⬖ is surjective.
INTERNAL DIRECT PRODUCTS
603
Also ᭙ x, ygG,
x y s Ž x ␣ 1 , x ␣ 2 . Ž y␣ 1 , y␣ 2 .  s Ž x ␣ 1 y␣ 1  , x ␣ 2 y␣ 2  .
2.1
s Ž xy ␣ 1 , xy ␣ 2 .
by Ž c .
s xy  .
⬖ is a homomorphism.
Finally let s y1 . Then G is an internal direct product as in Definition 2.2. Theorem 2.5 offers a useful alternative point of view. The class of all groupoid internal direct products may now be regarded as a variety of Ž  , ␣ 1 , ␣ 2 .-algebras, with the formulae 2.5Ža. ᎐ Žc. as axioms. ŽThese axioms are in fact independent.. Subsequent proofs refer back to 2.5Ža. ᎐ Žc. rather than directly to Definition 2.2. Between those general proofs there are also references to special properties commonly expected of internal direct products. The following list seems to cover most such properties. CONDITIONS 2.6. Custom prefers an internal direct product to satisfy: Ža. G has a neutral Židentity. element. Žb. Every element of G1 commutes with every element of G 2 . Žc. < G 2 ␣ 1 < s < G1 ␣ 2 < s 1. Žd. G 2 ␣ 1 s G1 ␣ 2 s a singleton invariant under ␣ 1 and ␣ 2 . Že. < G1 l G 2 < s 1. Žf. ␣ 1 < G 1 and ␣ 2 < G 2 are identity mappings. Equivalently, ␣ 12 s ␣ 1 and ␣ 22 s ␣ 2 . The statement of Žd. may appear awkward, especially since the singleton in Žd. and Že. traditionally contains just the identity element mentioned in Ža.. However the six statements Ža. ᎐ Žf. as they stand are useful for reference later. Rather obviously, they are not independent. Much less obviously, they can all be false. ŽCf. the last of Examples 7.8..
3. BASIC THEOREMS Examples 7.3᎐7.9 show that 2.6Žf. cannot be proven in general. However 2.5Žc. means that ␣ 1 and ␣ 2 must be endomorphisms, and Theorem 3.2 shows that their restrictions to G1 and G 2 , respectively, must be injective endomorphisms.
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First it is convenient to have notation for right and left translations. DEFINITION 3.1. For any a g G, ᭙ x g G, x a s xa  and x a s ax  . THEOREM 3.2. In any internal direct product Ž G,  , ␣ 1 , ␣ 2 . with G1 s G␣ 1 and G 2 s G␣ 2 , ␣ 1 < G 1 and ␣ 2 < G 2 are injections. Proof. Choose any a2 g G 2 Žwhich is non-empty by 2.2.. Then ᭙ x 1gG1 ,
x 1 ␣ 1 a 2 ␣ 1 s x 1 ␣ 1 a2 ␣ 1 
3.1
s x 1 a2 ␣ 1
2.5 Ž c .
s x1 .
2.5 Ž b .
Thus ␣ 1 < G 1 has a right inverse, so it is injective. Similarly ␣ 2 < G 2 is injective. COROLLARY. For any a2 g G 2 , a 2 ␣ 1 < G 1␣ 1 is a right in¨ erse of ␣ 1 < G 1. For any a1 g G1 , a1␣ 2 < G 2 ␣ 2 is a right in¨ erse of ␣ 2 < G 2 . In light of Theorems 2.5Žc. and 3.2, the question whether ␣ 1 < G 1 is an actual automorphism is simply the question whether G1 ␣ 1 s G1. Examples 7.5᎐7.9 show that this is not always so. But the condition G1 ␣ 1 s G1 is a significant restriction Žas is G 2 ␣ 2 s G 2 ., which the next theorem expresses in various forms. THEOREM 3.3. In any internal direct product Ž G,  , ␣ 1 , ␣ 2 . with G1 s G␣ 1 and G 2 s G␣ 2 , the following conditions are equi¨ alent. Ža. G1 ␣ 1 s G1. Žb. G 2 ␣ 1 is a singleton d14 with d1 ␣ 1 s d1. Žc. G1 ␣ 1 l G 2 ␣ 1 is a singleton d14 . Žd. G1 ␣ 1 l G 2 ␣ 1 / ⭋. Four other mutually equi¨ alent conditions are obtained by transposing the subscripts 1 and 2 abo¨ e. Proof. Ža. « Žb.. Choose any c1 g G1 Žwhich is non-empty.. Then c1 ␣ 2 ␣ 1 g G1 s G1 ␣ 1 from Ža., so ᭚d1 g G1: c1 ␣ 2 ␣ 1 s d1 ␣ 1. ⬖ ᭙ x 2gG 2 ,
x 2 ␣ 1 s c1 x 2 ␣ 2 ␣ 1 s c1 ␣ 2 ␣ 1 x 2 ␣ 2 ␣ 1 
2.5 Ž b . 2.5 Ž c .
s d1 ␣ 1 x 2 ␣ 2 ␣ 1  s d 1 x 2 ␣ 2 ␣ 1
2.5 Ž c .
s d1 ,
2.5 Ž b .
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INTERNAL DIRECT PRODUCTS
⬖G 2 ␣ 1 s d14 , so d1 is unique. Also d1 ␣ 1 s c1 ␣ 2 ␣ 1 g G 2 ␣ 1 s d14 . Hence Žb.. Žb. « Žc.. From Žb., G 2 ␣ 1 s d1 4 s d1 ␣ 1 4 : G1 ␣ 1. Hence Žc.. Žc. « Žd. obviously. Žd. « Ža.. From Žd., ᭚a1 g G1 , ᭚a2 g G 2 : a1 ␣ 1 s a2 ␣ 1. ⬖ ᭙ x 1gG1 ,
x 1 s x 1 a2 ␣ 1 s x 1 ␣ 1a 2 ␣ 1 
2.5 Ž b . 2.5 Ž c .
s x 1 ␣ 1a 1 ␣ 1  s x 1 a 1 ␣ 1
2.5 Ž c .
g G1 ␣ 1 . ⬖G1 : G1 ␣ 1. Also G1 ␣ 1 : G␣ 1 s G1. Hence Ža.. Similar proofs apply throughout when subscripts 1 and 2 are transposed. COROLLARY. Under the conditions 3.3Ža. ᎐ Žd., d1 < G 1 is the Ž two-sided. in¨ erse automorphism of ␣ 1 < G 1. Under the corresponding conditions with transposed subscripts, d 2 < G 2 is the Ž two-sided. in¨ erse automorphism of ␣2 < G2. Proof. Rewrite the corollary to Theorem 3.2 using 3.3Ža. and Žb., and also 2.5Žc.. Of the two statements in 3.3Žb., the invariance under ␣ 1 is the vital one. Condition 3.3Žb. can in fact be weakened to G 2 ␣ 12 s G 2 ␣ 1 with impunity, but Examples 7.5᎐7.7 show that it cannot be weakened merely to < G 2 ␣ 1 < s 1. Conditions 3.3Ža. ᎐ Žd. are not equivalent to the four conditions obtained from them by transposing subscripts. Examples 7.6 Žwith k G 1. satisfy the former but not the latter, and Examples 7.5 and 7.8 satisfy the latter but not the former. However the next theorem deals with the situation where both are satisfied simultaneously. THEOREM 3.4. In any internal direct product Ž G,  , ␣ 1 , ␣ 2 . with G1 s G␣ 1 and G 2 s G␣ 2 , the following conditions are equi¨ alent. Ža. G1 ␣ 1 s G1 and G 2 ␣ 2 s G 2 . Žb. G 2 ␣ 1 s G1 ␣ 2 s a singleton d4 with d ␣ 1 s d ␣ 2 s d. Žc. G1 l G 2 s a singleton d4 . Žd. G1 l G 2 / ⭋. Proof. Ža. « Žb.. From Ža., Theorem 3.3 gives G 2 ␣ 1 s d1 4 with d1 ␣ 1 s d1 , and G1 ␣ 2 s d 2 4 with d 2 ␣ 2 s d 2 . Also d 2 ␣ 1 g G 2 ␣ 1 s
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K. E. PLEDGER
d14 , so d 2 ␣ 1 s d1. Similarly d1 ␣ 2 s d 2 . ⬖ d1 s d 1 ␣ 1 d 1 ␣ 2 
2.5 Ž a .
s d1 d 2  s d 2 ␣ 1 d 2 ␣ 2  s d 2
2.5 Ž a .
s d, say. Hence Žb.. Žb. « Žc.. From Žb., d g G 2 ␣ 1 l G1 ␣ 2 : G1 l G 2 . Also ᭙ x g G1 l G 2 , x ␣ 1 g Ž G1 l G 2 . ␣ 1 : G 2 ␣ 1 s d4 and x ␣ 2 g Ž G1 l G 2 . ␣ 2 : G1 ␣ 2 s d 4 , so x s x ␣1 x ␣2 
2.5 Ž a .
s dd  s d
⬗ d 4 is a subgroupoid.
Hence Žc.. Žc. « Žd. obviously. Žd. « Ža.. Ž G1 l G 2 . ␣ 1 : G1 ␣ 1 l G 2 ␣ 1. Hence G1 l G 2 / ⭋
«
G1 ␣ 1 l G 2 ␣ 1 / ⭋
«
G1 ␣ 1 s G1
3.3
and similarly G2 ␣ 2 s G2 .
COROLLARY 1. Under the conditions 3.4Ža. ᎐ Žd., d < G 1 and ␣ 1 < G 1 are in¨ erse automorphisms, d < G 2 and ␣ 2 < G 2 are in¨ erse automorphisms, and dd  s d. Proof. In the corollary to Theorem 3.3, put d1 s d 2 s d. Also observe that d4 is a subgroupoid. COROLLARY 2.
< G1 l G 2 < can only be 1 or 0.
Proof. 3.4Žc. m 3.4Žd.. Cases with < G1 l G 2 < s 1 satisfy all the Theorem 3.4 conditions and clearly generalize traditional internal direct products, whereas cases with < G1 l G 2 < s 0 satisfy none of the Theorem 3.4 conditions and are much more novel. Examples 7.5᎐7.9 Žand Section 8. exhibit some of the tangled consequences of G1 l G 2 s ⭋. In the meantime Theorems 3.5᎐6.6 explore the tidier situation where G1 l G 2 s d4 .
INTERNAL DIRECT PRODUCTS
607
First, Theorem 3.4 Corollary 1 has something like a converse. THEOREM 3.5. If a groupoid Ž G1 ,  . has an idempotent element d1 such that d1 is an automorphism, then G1 is isomorphic to the left factor of an internal direct product. If a groupoid Ž G 2 ,  . has an idempotent element d 2 such that d 2 is an automorphism, then G 2 is isomorphic to the right factor of an internal direct product. Any such G1 can be combined with any such G 2 , and the resulting internal direct product satisfies the Theorem 3.4 conditions. Proof. Both G1 and G 2 will be needed in the construction. So, if given only G1 , let G 2 be the trivial group d 2 4 ; if given only G 2 , let G1 be the trivial group d14 . Then in all cases both G1 and G 2 are available. d1 is an automorphism of G1 , so d1y1 is also. Likewise d 2y1 is an automorphism of G 2 . Hence we can define within G1 = G 2
Ž x 1 , x 2 . ␣ 1 s ž x 1 d1y1 , d 2 / , and
Ž x 1 , x 2 . ␣ 2 s ž d1 , x 2 d 2y1 / . The verification of 2.5Ža. ᎐ Žc. is then quite routine, and is omitted here. Also
Ž G1 = G 2 . ␣ 1 s G1 = d 2 4 ( G1
⬗ d1y1 is an automorphism of G1 ⬗ d 2 is idempotent,
and similarly Ž G1 = G 2 . ␣ 2 s d14 = G 2 ( G 2 . Thus Ž G1 = G 2 ,  . is an internal direct product of subgroupoids isomorphic to G1 and G 2 . These subgroupoids have intersection Ž G1 = d 2 4. l Ž d14 = G 2 . s Ž d1 , d 2 .4 as in Theorem 3.4Žc.. The main difficulty in applying Theorem 3.5 is notational. In practice the construction is easier if the embeddings of G1 and G 2 are treated as identifications. For this write d1 s d 2 s d, choose notation to make G1 l G 2 s d4 only, and write the elements of G as x 1 x 2  rather than Ž x 1 , x 2 .. Then G1 and G 2 are themselves subgroupoids of G, as in Examples 7.2᎐7.4. The idempotence condition dd  s d is vital to the Theorem 3.5 construction. In Example 7.1, both a and b are automorphisms of G1 , and both f and g are automorphisms of G 2 ; but the direct product cannot be made internal as above, because none of a, b, f, and g is idempotent.
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K. E. PLEDGER
4. ANALYZING THE TRADITION It is now possible to look in more detail at the traditional properties 2.6Ža. ᎐ Žf. and relations between them. Theorem 3.4 already includes 2.6Žd. and Že., and the following theorems treat internal direct products which are closer to tradition in various other ways. The first of them involves 2.6Že. and Žb.. THEOREM 4.1. If G1 l G 2 s d4 , and d commutes with e¨ ery element of G, then e¨ ery element of G1 commutes with e¨ ery element of G 2 . Proof. ᭙ x 1 g G1 , ᭙ x 2 g G 2 , x 2 x 1  s dx 2 ␣ 2  x 1 ␣ 1 d 
2.5 Ž a . , 3.4 Corollary 1
s dx 1 ␣ 1  x 2 ␣ 2 d 
2.4
s x 1 ␣ 1 d  dx 2 ␣ 2 
if d commutes with every element
s x1 x 2  .
2.5 Ž a . , 3.4 Corollary 1
Ž ⬗ d g G1 and d g G2 .
This generalizes the traditional proof which assumes an identity element and simply substitutes it for x 1 and y 2 in Theorem 2.4. The converse of Theorem 4.1 is false. Example 7.6 Žwith k s 1. has every element of G1 commuting with every element of G 2 , but G1 l G 2 empty. Now, 2.6Žf. includes the claim that ␣ 1 < G 1 is an identity mapping. Theorem 3.3 and its corollary give conditions merely for ␣ 1 < G 1 to be an automorphism, but by an easy deduction it is idempotent and hence the identity mapping of G1 if and only if G 2 ␣ 1 s d14 where d1 is a right identity element for G1. Rather than spend time on that and on the corresponding result for ␣ 2 , the next theorem strengthens Theorem 3.4 Žinvolving both ␣ 1 and ␣ 2 . to the whole of 2.6Žf.. THEOREM 4.2. ␣ 1 < G 1 and ␣ 2 < G 2 are both identity mappings Ž i.e., ␣ 12 s ␣ 1 and ␣ 22 s ␣ 2 . if and only if some element d in G1 l G 2 is both a right identity element for G1 and a left identity element for G 2 . In such a case d4 s G 1 l G 2 s G 2 ␣ 1 s G 1 ␣ 2 . Proof. Assume both ␣ 1 < G 1 s 1 G 1 and ␣ 2 < G 2 s 1 G 2 . This is a special case of 3.4Ža.. Hence 3.4Žc. and Žb. give d4 s G1 l G 2 s G 2 ␣ 1 s G1 ␣ 2 , and 3.4 Corollary 1 gives d < G 1 s 1 G 1 and d < G 2 s 1 G 2 so d has the required identity properties.
INTERNAL DIRECT PRODUCTS
609
Conversely, assume an element d in G1 l G 2 is both a right identity element for G1 and a left identity element for G 2 . Then 3.4Žd. holds, so 3.4Žc. and Žb. give d4 s G1 l G 2 s G 2 ␣ 1 s G1 ␣ 2 , and 3.4 Corollary 1 gives ␣ 1 < G 1 s 1 G 1 and ␣ 2 < G 2 s 1 G 2 . Łos´ included 2.6Žf. in his definition of internal direct product w6, p. 34, Ž1.x, but he missed the consequence that G1 l G 2 / ⭋. ŽCf. w6, p. 35, Lemma 3x.. An effect of combining 2.6Žb. and Žf. can be inferred from 4.2 as THEOREM 4.3. If e¨ ery element of G1 commutes with e¨ ery element of G 2 , and ␣ 12 s ␣ 1 and ␣ 22 s ␣ 2 , then G has an identity element. Proof. Assuming ␣ 12 s ␣ 1 and ␣ 22 s ␣ 2 , apply Theorem 4.2 but write e instead of d. Then in particular e g G1 , and e is a left identity element for G 2 . Thus, assuming every element of G1 commutes with every element of G 2 , e is also a right identity element for G 2 . ⬖ ᭙ xgG,
xe  sx ␣ 1 x ␣ 2  ee 
2.5 Ž a .
s x ␣ 1 e  x ␣ 2 e 
2.4, ⬗ e g G1 and e g G 2
s x ␣1 x ␣2 
⬗ e is a right identity for G1 and G 2
sx
2.5 Ž a .
s ex 
similarly.
The last of these miscellaneous theorems combines 2.6Ža. and Že.. THEOREM 4.4. d s e.
If G has identity element e, and G1 l G 2 s d4 , then
Proof. Assume G1 l G 2 s d4 as in 3.4Žc.. Then 3.4Žb. gives d s d ␣ 1 s d ␣ 2 , so d ␣ 1 s d ␣ 12 s ed ␣ 12 s e ␣ 1 d ␣ 1 ␣ 1
assuming identity element e 2.5 Ž c .
s e ␣ 1 d ␣ 2 ␣ 1 s e␣1 ,
2.5 Ž b .
and similarly d ␣ 2 s e ␣ 2 . Combine these two equations using 2.5Ža..
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K. E. PLEDGER
Theorem 4.4 does not assert that if G has identity element e, then G1 l G 2 s e4 : the hypothesis G1 l G 2 s d4 is vital. Some of Examples 7.8 show that when G1 l G 2 is empty, G may yet have an identity element somewhere else. Implications connecting 2.6Ža. ᎐ Žf. can now be assembled. Theorem 4.2 shows that Žf. « Že.. Theorem 3.4 shows that Že. m Žd.. Trivially Žd. « Žc.. Theorems 4.4 and 4.1 show that Ža. n Že. « Žb.. Theorems 4.4 and 4.2 show that Ža. n Že. « Žf.. Theorem 4.3 shows that Žb. n Žf. « Ža.. From all that, the appropriate Boolean disentangling sorts internal direct products into 12 classes partially ordered by inclusion as in Fig. 4.5. ŽFor simplicity, one-sided conditions such as in Theorem 3.3 are not included here.. The figure’s notation Žc, d, e., for example, means the class of all internal direct products satisfying 2.6Žc., Žd., and Že.. The 12 classes in the figure are all distinct. Example 7.2 satisfies 2.6Žc᎐f. but neither Ža. nor Žb., so every class containing Žc, d, e, f. is distinct from every class contained in Ža. or Žb.. Example 7.3 shows likewise that every class containing Žb, c, d, e. is distinct from every class contained in Ža. or Žc, d, e, f.. Various cases of Example 7.8 show that every class containing Ža, b, c. is distinct from every class contained in Žc, d, e., every class containing Ža, b. is distinct from every class contained in Žc., and every class containing Ža, c. is distinct from every class contained in Žb.. Perhaps Theorem 3.4 Corollary 2 locates the major watershed, between internal direct products having < G1 l G 2 < s 1 and those having < G1 l G 2 < s 0, i.e., between members and non-members of Žc, d, e.. Within that class, Fig. 4.5 shows that Žb. and Žf. Žessentially as in Theorems 4.1 and 4.2. may be regarded as two independent components which together strengthen the condition < G1 l G 2 < s 1 to the fully classical Ža, b, c, d, e, f..
Ž All internal direct products. Ža.
Žb.
Žc.
Ža, c. Ža, b.
Žc, d, e. Žb, c.
Ža, b, c.
Žb, c, d, e. Ža, b, c, d, e, f.
FIGURE 4.5
Žc, d, e, f.
INTERNAL DIRECT PRODUCTS
611
5. STANDARD SPECIAL PROPERTIES Some familiar groupoid properties force internal direct products to satisfy 2.6Žc, d, e. or even Žc, d, e, f.. The next three theorems show why. As usual G / ⭋ from Definition 2.2. THEOREM 5.1.
If G is finite, then < G1 l G 2 < s 1.
Proof. By Theorem 3.2, ␣ 1 < G 1 and ␣ 2 < G 2 are injective. Hence in the finite case G1 ␣ 1 s G1 and G 2 ␣ 2 s G 2 . The result now follows from Theorem 3.4. THEOREM 5.2. < G1 l G 2 < s 1.
If G satisfies either the right or left cancellation law, then
Proof. Assume G satisfies the left cancellation law. Choose any a1 g G1 and a2 g G 2 . Then a1 ␣ 1 a1 ␣ 2  s a1
2.5 Ž a .
s a1 a2 ␣ 1
2.5 Ž b .
s a1 ␣ 1 a2 ␣ 1  .
2.5 Ž c .
Cancel a1 ␣ 1 from the left. ⬖ a1 ␣ 2 s a2 ␣ 1. ⬖ G1 l G 2 / ⭋. The result now follows from Theorem 3.4. The proof is similar if G has right cancellation. It may be observed that the foregoing proof really requires only one left cancellable element in G1 ␣ 1 or one right cancellable element in G 2 ␣ 2 . The last two theorems show that either finiteness or a cancellation law puts an internal direct product into the class Žc, d, e. of Fig. 4.5. However, it need not be in either of the subclasses Žb, c, d, e. or Žc, d, e, f.: Example 7.4 is a finite cancellation groupoid in which d is neither central Žas in 4.1. nor a one-sided identity element for G1 or G 2 Žas in 4.2.. The next result is stronger: the associative law puts an internal direct product into the class Žc, d, e, f.. In fact 2.6Žf. is equivalent to a restricted associative condition, as follows. THEOREM 5.3. ᭙ x 1 g G1 ,
␣ 12 s ␣ 1 and ␣ 22 s ␣ 2 if and only if ᭙ y g G,
᭙ z 2 g G2 ,
x 1 y  z 2  s x 1 yz 2  .
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K. E. PLEDGER
Proof. Assume every x 1 y  z 2  s x 1 yz 2  . Choose any a2 , b 2 g G 2 . Then ᭙ x 1gG1 ,
x 1 ␣ 1 s x 1 ␣ 1 a2 b 2 ␣ 1
2.5 Ž b .
s x 1 ␣ 1 a2  b 2 ␣ 1
from the above assumption
s x 1 ␣ 1 a2 ␣ 1 b 2 ␣ 1 
2.5 Ž c .
s x1 ␣ 1 b2 ␣ 1 
2.5 Ž b .
s x 1 b 2 ␣ 1
2.5 Ž c .
s x1 .
2.5 Ž b .
⬖ ␣ 1 < G 1 s 1 G 1. Similarly ␣ 2 < G 2 s 1 G 2 . Conversely, assume ␣ 12 s ␣ 1 and ␣ 22 s ␣ 2 . Then an element d has properties as in Theorem 4.2. Now, for every x 1 g G1 and y g G, x 1 y ␣ 1 s x 1 ␣ 1 y␣ 1  ⬖ x 1 y ␣ 1 d  s x 1 ␣ 1 y␣ 1 d 
2.5 Ž c . ⬗ d is a right identity for G1 , by 4.2.
Hence, for every z 2 g G 2 , x 1 y ␣ 1 z 2 ␣ 1  s x 1 ␣ 1 y␣ 1 z 2 ␣ 1 
⬗ G 2 ␣ 1 s d 4 by 4.2
⬖ x 1 y  z 2 ␣ 1 s x 1 ␣ 1 yz 2 ␣ 1 
2.5 Ž c .
s x 1 yz 2 ␣ 1 .
2.5 Ž c .
Similarly x 1 y  z 2 ␣ 2 s x 1 yz 2 ␣ 2 . Combine the last two equations using 2.5Ža.. ⬖ x 1 y  z 2  s x 1 yz 2  . COROLLARY. If G is a semigroup, then ␣ 12 s ␣ 1 , ␣ 22 s ␣ 2 , and G1 l G 2 s d4 where d is a right identity element for G1 and a left identity element for G 2 . Proof. The equation in Theorem 5.3 is a restricted case of the associative law. The result follows from Theorems 5.3 and 4.2. The corollary shows that the associative law puts an internal direct product into the class Žc, d, e, f. of Fig. 4.5. However it need not be in the subclass Ža, b, c, d, e, f.: Example 7.2 is a semigroup in which d is neither an identity element Žas in 4.4. nor even central Žas in 4.1.. Theorems 5.1᎐5.3 ensure that most well-known groupoids can form internal direct products only with < G1 l G 2 < s 1. Any example with < G1 l G 2 < s 0 must be infinite and without the associative law or either cancellation law. ŽPerhaps this helps to explain why Examples 7.5᎐7.9 may
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INTERNAL DIRECT PRODUCTS
seem far-fetched.. However, the commutative law does not force < G1 l G 2 < to be 1: Example 7.6 Žwith k s 1. is commutative but has G1 l G 2 empty. Also the condition < G1 l G 2 < s 1 does not enforce any of the special properties of Theorems 5.1᎐5.3: Example 7.6 Žwith k s 0. has < G1 l G 2 < s 1 but enjoys neither finiteness nor the associative nor cancellation law.
6. CLOSER TO TRADITION Consider any internal direct product G having < G1 l G 2 < s 1, and hence belonging to the class Žc, d, e. of Fig. 4.5. Although G need not satisfy 2.6Žf. and the corresponding properties in 4.2, it is always isotopic to a groupoid which has those properties and so belongs to the class Žc, d, e, f.. If G is in Žb, c, d, e., then the isotopic groupoid is in Ža, b, c, d, e, f.. Several construction steps lead to the main result in Theorem 6.6. DEFINITIONS 6.1 Žof ␥ 11 , ␦ , 1 , 2 .. ᭙ x g G, x␥ 11 s x ␣ 12 x ␣ 22 . The binary operation ␦ s ␥ 11 . When G1 l G 2 s d4 , then ᭙ x g G, x 1 s x ␣ 1 d  , and x 2 s dx ␣ 2  . The notation ␥ 11 is part of a systematic notation ␥m n mentioned in Section 8. Theorem 6.6 shows that Ž G, ␦ , 1 , 2 . is an internal direct product of G1 and G 2 , satisfying 2.6Žc᎐f., and having Ž G, ␦ . isotopic to Ž G,  .. The proof uses several lemmas. LEMMA 6.2. ␥ 11 is an endomorphism of Ž G,  .. Proof. ᭙ x, ygG,
xy ␥ 11 s xy ␣ 12 xy ␣ 22
6.1
s x ␣ 12 y␣ 12 x ␣ 22 y␣ 22
2.5 Ž c .
s x ␣ 12 x ␣ 22 y␣ 12 y␣ 22
2.4
s x␥ 11 y␥ 11  .
6.1
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K. E. PLEDGER
LEMMA 6.3.
If G1 l G 2 s d4 , then ␥ 11 1 s 1 ␥ 11 s ␣ 1 , ␥ 11 2 s 2 ␥ 11 s ␣ 2 , G 1 s G1 ,
and
G 2 s G 2 .
Proof. ᭙ xgG,
᭙ xgG,
x␥ 11 1 s x ␣ 12 x ␣ 22␣ 1 d 
6.1
s x ␣ 12 d 
2.5 Ž b .
s x ␣1
3.4 Corollary 1
x 1 ␥ 11 s x ␣ 1 d ␣ 12 x ␣ 1 d ␣ 22
6.1
s x ␣1 d␣2 
2.5 Ž b . , ⬗ d g G 2
s x ␣ 12 d 
3.4 Ž b .
s x ␣1 .
3.4 Corollary 1
2
Similarly ␥ 11 2 s 2 ␥ 11 s ␣ 2 . ᭙ xgG,
x ␣ 1 s x␥ 11 1
from above
g G 1 . ⬖ G1 : G 1. ᭙ xgG,
x 1 s x ␣1 d g G1 .
6.1 ⬗ d g G1
⬖ G 1 : G1. ⬖ G 1 s G1. Similarly G 2 s G 2 . LEMMA 6.4. product.
If G1 l G 2 s d4 , then Ž G, ␦ , 1 , 2 . is an internal direct
Proof. The appropriate versions of 2.5Ža. ᎐ Žc. are verified as follows.
Ž a.
Ž b.
᭙ xgG,
᭙ x, ygG,
x 1 x 2 ␦ s x 1 x 2 ␥ 11
6.1
s x 1 ␥ 11 x 2 ␥ 11 
6.2
s x ␣1 x ␣2 
6.3
s x.
2.5 Ž a .
x 1 y 2 ␦ 1 s x 1 y 2 ␥ 11 1
6.1
s x 1 y 2 ␣ 1
6.3
s x1;
2.5 Ž b .
since 6.3 shows x 1 g G1 and y 2 g G 2 .
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INTERNAL DIRECT PRODUCTS
Similarly x 1 y 2 ␦ 2 s y 2 . Žc. For every x, y g G, xy ␣ 1 s x ␣ 1 y␣ 1  ⬖ xy ␥ 11 1 s x 1 ␥ 11 y 1 ␥ 11  s x 1 y 1 ␥ 11 ⬖ xy␦ 1 s x 1 y 1 ␦ .
2.5 Ž c . 6.3 6.2 6.1
Similarly xy␦ 2 s x 2 y 2 ␦ . LEMMA 6.5. If G1 l G 2 s d4 , then ␥ 11 is an automorphism of Ž G,  ., ha¨ ing in¨ erse x ¬ x 1 x 2  . Proof. ᭙ xgG,
x 1 x 2 ␥ 11 s x 1 x 2 ␦
6.1
s x,
6.4 Ž a .
and x␥ 11 1 x␥ 11 2  s x ␣ 1 x ␣ 2 
6.3
s x.
2.5 Ž a .
y1
Thus x␥ 11 s x 1 x 2  , so the endomorphism ␥ 11 of Lemma 6.2 is in fact an automorphism. Lemma 6.2 does not require that G1 l G 2 / ⭋, but Lemma 6.5 does: cf. Examples 7.5᎐7.9 and Section 8. ŽIn fact the converse of Lemma 6.5 can be proven.. THEOREM 6.6. Any internal direct product Ž G,  , ␣ 1 , ␣ 2 . with G1 l G 2 s d4 has the following properties. Ža. Ž G,  . is isotopic to Ž G, ␦ .. Žb. Ž G, ␦ , 1 , 2 . is an internal direct product of G1 and G 2 . Žc. 12 s 1 and 22 s 2 . Žd. d is a right identity element for Ž G1 , ␦ . and a left identity element for Ž G 2 , ␦ .. Proof.
Ž a.
␥ 11 is a permutation of G
6.5
and ᭙ x, y g G,
xy ␥ 11 s xy␦ .
6.1
⬖ Ž1 G , 1 G , ␥ 11 . is an isotopism from Ž G,  . onto Ž G, ␦ .. ŽLemma 6.5 shows that Ž␥ 11 , ␥ 11 , 1 G . is another..
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K. E. PLEDGER
Žb. Lemma 6.3 shows G 1 s G1 follows from Lemma 6.4.
Ž c . ᭙ xgG,
G 2 s G 2 . The result then
and
x 12 s x ␣ 1 d ␣ 1 d 
6.1
s x ␣1 d
2.5 Ž b . , ⬗ d g G2
s x 1.
6.1
Similarly 22 s 2 . Žd. now follows from Theorem 4.2.
7. EXAMPLES In each of the finite examples 7.1᎐7.4, G 2 is the opposite of G1. The infinite examples begin with 7.5. EXAMPLE 7.1. An external direct product which cannot be made internal. G1
a
b
G2
f
g
a b
b a
b a
f g
g g
f f
G1 = G 2
Ž a, f .
Ž a, g .
Ž b, f .
Ž b, g .
Ž a, f . Ž a, g . Ž b, f . Ž b, g .
Ž b, g . Ž b, g . Ž a, g . Ž a, g .
Ž b, f . Ž b, f . Ž a, f . Ž a, f .
Ž b, g . Ž b, g . Ž a, g . Ž a, g .
Ž b, f . Ž b, f . Ž a, f . Ž a, f .
G1 = G 2 has no non-trivial subgroupoids at all, so it is certainly not an internal direct product of two subgroupoids. Section 1 referred to this difficulty. As mentioned at the end of Section 3,
a s b s the transposition Ž a b . s an automorphism of G1 and
f s g s the transposition Ž f g . s an automorphism ofG 2 . However none of a, b, f, g is idempotent, so Theorem 3.5 does not apply.
INTERNAL DIRECT PRODUCTS
617
EXAMPLE 7.2. A semigroup which is an internal direct product satisfying 2.6 Žc, d, e, f. but neither Ža. nor Žb.. G1
d
a
G2
d
f
d a
d a
d a
d f
d d
f f
Here dd  s d, d is the identity automorphism of G1 , and d is the identity automorphism of G 2 , so the 3.5 conditions are satisfied. Hence combine G1 and G 2 as described after Theorem 3.5, writing t for af  . That gives: x x ␣1 x ␣2
d d d
a a d
f d f
t a f
Thus, for example, tf ␣ 1 s t ␣ 1 f ␣ 1 
2.5 Ž c .
s ad  s a
from tables above
and tf ␣ 2 s t ␣ 2 f ␣ 2  s ff  s f so tf  s tf ␣ 1 tf ␣ 2 
2.5 Ž a .
s af  s t. Calculating by components in that way leads to this table: G
d
a
f
t
d a f t
d a d a
d a d a
f t f t
f t f t
Of course the last two tables could have been given in the first place, but then they would need to be checked for conditions 2.5Ža. ᎐ Žc.. The G1 and G 2 tables occur within the G table. The table for ␣ 1 and ␣ 2 attests 2.6Žf.: ␣ 1 < G 1 and ␣ 2 < G 2 are identity mappings. Theorem 4.2 restates this in terms of d. Indeed, G1 has d Žand
618
K. E. PLEDGER
also a. as a right but not left identity element, and G 2 has d Žand also f . as a left but not right identity element. But neither d nor anything else is even a one-sided identity element for G itself, so 2.6Ža. is false. Also a g G1 and d g G 2 but a does not commute with d, so 2.6Žb. is false. Since G1 and G 2 are obviously associative, G is also. Finally, G1 alone Žor G 2 alone. illustrates the introductory paragraph of the proof of Theorem 3.5. Since dd  s d and d is an automorphism of G1 , the groupoid G1 can be seen as the internal direct product of itself and d4 . EXAMPLE 7.3. A commutative quasigroup which is an internal direct product satisfying 2.6Žb, c, d, e. but neither Ža. nor Žf..
Ž G1 ,  .
d
a
b
Ž G2 ,  .
d
f
g
d a b
d b a
b a d
a d b
d f g
d g f
g f d
f d g
Here d < G 1 is the transposition Ž a b . which is easily seen to be an automorphism of G1. Likewise d < G 2 is an automorphism of G 2 , and also dd  s d. Hence Theorem 3.5 applies, so G1 and G 2 can be combined as in the previous example to give a groupoid G of order 9, not tabulated here. By Theorem 3.4 Corollary 1, ␣ 1 < G 1 is the inverse of d < G 1 so it also is Ž a b ., and similarly ␣ 2 < G 2 s Ž f g .. Now, d commutes with every element, so Theorem 4.1 ensures 2.6Žb.: every element of G1 commutes with every element of G 2 . As an example of the Theorem 4.1 proof, ga  s df  bd  s db fd  s bd  df  s ag  . But neither d nor anything else is even a one-sided identity element, so 2.6Ža. is false. Also ␣ 1 < G 1 and ␣ 2 < G 2 as given above are not identity mappings, so 2.6Žf. is false. Incidentally, G is a finite commutative quasigroup with G1 ( G 2 , but nevertheless it satisfies neither 2.6Ža. nor Žf.. This is a convenient example to illustrate Section 6. Using Definition 6.1, ␥ 11 turns out to be the permutation Ž a b .Ž f g .Ž af  bg  .Ž ag  bf  . leaving d invariant. Hence ␦ s ␥ 11 gives, in particular, new tables for G1 and G 2 :
Ž G1 , ␦ .
d
a
b
Ž G2 , ␦ .
d
f
g
d a b
d a b
a b d
b d a
d f g
d f g
f g d
g d f
619
INTERNAL DIRECT PRODUCTS
These are both groups, so their direct product Ž G, ␦ . is just the elementary Abelian group of order 9. Other details from Theorem 6.6 are easy to check. EXAMPLE 7.4. A finite quasigroup which is an internal direct product satisfying 2.6Žc, d, e. but neither Ža., Žb., nor Žf.. G1
d
a
b
c
G2
d
f
g
h
d a b c
d b c a
c a d b
a c b d
b d a c
d f g h
d h f g
g f h d
h d g f
f g d h
Here d < G 1 is the cycle Ž a b c . which is an automorphism of G1. Likewise d < G 2 is an automorphism of G 2 , and also dd  s d. Hence Theorem 3.5 applies, so G1 and G 2 can be combined as in Example 7.2 to give a groupoid G of order 16, not tabulated here. By Theorem 3.4 Corollary 1, ␣ 1 < G 1 is the inverse of d < G 1 so it is Ž a c b ., and similarly ␣ 2 < G 2 s Ž f h g .. G is a finite cancellation groupoid Žbecause both G1 and G 2 are., and indeed satisfies 2.6Žc᎐e. as Theorems 5.1 and 5.2 require. But it has not even a one-sided identity element, so 2.6Ža. is false. Also d commutes with no other element, so 2.6Žb. is false. Also ␣ 1 < G 1 and ␣ 2 < G 2 as given above are not identity mappings, so 2.6Žf. is false. Like the previous example, this one can be used to illustrate Theorem 6.6, although in this case Ž G, ␦ . is not a group. EXAMPLES 7.5. The simplest internal direct products having G1 l G 2 s ⭋, i.e., not satisfying 2.6Žc, d, e.. Theorem 5.1 shows that any such G must be infinite, so at least one of G1 and G 2 is infinite. Hence the smallest possible orders < G1 < and < G 2 < are / 0 and 1 Žor vice versa., as in these examples. G s ⺞ j 0 4 s 0, 1, 2, . . . 4 . ᭙ m, n g G,
¡ ¢
my1 mn  s m m q Ž n y m.
~
if m ) n if m s n, if m - n
where is any mapping ⺞ ª ⺞ j 04 . ŽDifferent choices of give different examples.. ᭙ n g G,
n ␣ 1 s n q 1 and
n ␣ 2 s 0.
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K. E. PLEDGER
Since cases with G1 l G 2 s ⭋ are unfamiliar, 2.5Ža. ᎐ Žc. are here verified in full.
Ž a. ᭙ ngG,
n ␣ 1 n ␣ 2  s Ž n q 1. 0  s Ž n q 1. y 1
⬗ nq1)0
s n. Žb. ᭙ m, n g G,
m ␣ 1 n ␣ 2 ␣ 1 s Ž m q 1 . 0 ␣ 1 s m ␣ 1
and m ␣ 1 n ␣ 2 ␣ 2 s 0 s n ␣ 2 .
Ž c.
m)n
«
m ␣ 1 n ␣ 1  s Ž m q 1. Ž n q 1.  s m s Ž m y 1 . ␣ 1 s mn ␣ 1
msn
«
m ␣ 1 n ␣ 1  s m ␣ 1 s mn ␣ 1
m-n
«
m ␣ 1 n ␣ 1  s Ž m q 1. Ž n q 1.  s m q 1 q Ž n y m . s mn  q 1 s mn ␣ 1
Also ᭙ m, n g G, m ␣ 2 n ␣ 2  s 00  s 0 s mn ␣ 2 . Hence G is indeed an internal direct product with G1 s G␣ 1 s ⺞ and G 2 s G␣ 2 s 04 , so < G1 < s / 0 and < G 2 < s 1. Incidentally also G1 j G 2 s G, showing another sense in which these examples are small. Clearly G1 l G 2 s ⭋, so 2.6Že. is false. In fact G is easily seen to satisfy 2.6Žc. but neither Ža., Žd., Že., nor Žf.. It satisfies 2.6Žb. if and only if is the mapping n ¬ n y 1 Žas in Example 7.6 with k s 1.. It follows from Theorems 5.1᎐5.3 that G is neither finite nor a cancellation groupoid nor a semigroup. Indeed, 00  s 0 s 10  and 20  s 1 s 21  are counter-examples to the two cancellation laws, and 21  0  s 10  s 0 / 1 s 20  s 210  is a counter-example to the associative law. Since G1 ␣ 1 s 2, 3, 4, . . . 4 / G1 , the conditions 3.3Ža. ᎐ Žd. are not satisfied. However, they are satisfied when the subscripts 1 and 2 are transposed, since G 2 ␣ 2 s 04 s G 2 . Section 6 mentioned that although Lemma 6.2 showed ␥ 11 to be an endomorphism in all cases, Lemma 6.5 only showed it to be an automorphism in cases where G1 l G 2 / ⭋. In the present examples, n␥ 11 s n ␣ 12 n ␣ 22 s Ž n q 2.0  s n q 1, so indeed ␥ 11 is not surjective. ŽSimilar remarks apply in Examples 7.6᎐7.9.. Theorem 2.5Žc. shows that ␣ 1 is an endomorphism, and in these examples ␣ 1: n ¬ n q 1 is injective. Hence G␣ 1 , G␣ 12 , etc. are subgroupoids isomorphic to G itself, and thus internal direct products in their
621
INTERNAL DIRECT PRODUCTS
own right. In particular, G␣ 1 s 1, 2, 3, 4, . . . 4 is an internal direct product of G␣ 12 s 2, 3, 4, . . . 4 and 14 . Thus G s internal direct product of G␣ 1 and G␣ 2 s internal direct product of Ž internal direct product of G␣ 12 and 1 4 . and 0 4 . But this internal direct multiplication is not associative. There is no internal direct product of 14 and 04 ; since their external direct product has order one, whereas there are two elements to accommodate. Hence any extension of internal direct products to more than two factors requires special care. EXAMPLES 7.6. Groupoids from 7.5 which factorize as internal direct products in more than one way. With the same Ž G,  . as in Examples 7.5, can the projections ␣ 1 and ␣ 2 be different from before? This can happen if Žand in fact only if. there is some integer k G 0 such that ᭙ n G MaxŽ k, 1., n s n y k. ŽIf k G 2, the values 1 , 2 , . . . , Ž k y 1. are still arbitrary non-negative integers.. Then Ž G,  . is an internal direct product not only as in 7.5 but also in a new way where ᭙ n g G, n ␣ 1 s 0, and n ␣ 2 s n q k. The verification of 2.5Ža. ᎐ Žc. is omitted this time. G1 s G␣ 1 s 04 and G 2 s G␣ 2 s k, k q 1, k q 2, . . . 4 . Case k s 0. ᭙ n g ⺞,
n s n, so mn  s
᭙ n g G,
n ␣ 1 s 0 and
½
my1 n
if m ) n if m F n
n ␣ 2 s n.
G is the internal direct product of G1 s 04 and G 2 s G s ⺞ j 04 . Hence G1 l G 2 s 04 / ⭋, and indeed G satisfies all of 2.6Žc᎐f. as in Theorem 4.2. Observe then that a groupoid may factorize as an internal direct product in one way which satisfies 2.6Že. as well as in another way Žcf. 7.5. which does not. Case k s 1. ᭙ n g ⺞,
n s n y 1, so mn  s
᭙ n g G,
n ␣ 1 s 0 and
½
my1 m ny1
if m ) n if m s n if m - n
n ␣ 2 s n q 1.
G is the internal direct product of G1 s 04 and G 2 s ⺞.
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K. E. PLEDGER
The formula for mn  shows that G is commutative, so the commutative law does not force G1 l G 2 to be non-empty Žas mentioned near the end of Section 5.. In particular, every element of G1 commutes with every element of G 2 , and indeed G satisfies 2.6Žb, c. but neither Ža., Žd., Že., nor Žf., so the converse of Theorem 4.1 is false. The present factorization of G as an internal direct product is just the 7.5 factorization with the subscripts 1 and 2 transposed. The commutative law makes this possible. Cases k G 1 have G1 ␣ 1 s 04 s G1 and G 2 ␣ 2 s 2 k, 2 k q 1, 2 k q 2, . . . 4 / G 2 , so they satisfy the conditions 3.3Ža. ᎐ Žd. but not the same conditions with the subscripts 1 and 2 transposed. Cases k G 2 have G / G1 j G 2 . EXAMPLES 7.7. Groupoids resembling Examples 7.5 but with G1 and G 2 both infinite. G s Ž ⺞ j 04 . = Ž ⺞ j 04 . . ᭙ m, n, p, q g ⺞ j 04 , Ž m, n.Ž p, q .  is the ordered pair whose first component
¡m y 1 s~m ¢m q Ž p y m.
and second component
¡q y 1 ¢q q Ž n y q .
s~q
2
1
if m ) p if m s p if m - p
if q ) n if q s n if q - n,
where 1 and 2 are any mappings ⺞ ª ⺞ j 04 . ŽDifferent choices of 1 and 2 give different examples.. ᭙ m, n g ⺞ j 04 ,
Ž m, n . ␣ 1 s Ž m q 1, 0 . and Ž m, n . ␣ 2 s Ž 0, n q 1 . . Verification of 2.5Ža. ᎐ Žc. is no harder than in Examples 7.5. G1 s G␣ 1 s ⺞ = 04 and G 2 s G␣ 2 s 04 = ⺞, so G1 l G 2 s ⭋. Indeed G is easily seen to satisfy 2.6Žc. but neither Ža., Žd., Že., nor Žf.. It satisfies 2.6Žb. if and only if ᭙ n g ⺞, n 1 s n 2 s n y 1, in which case G is commutative. It violates both parts of 3.4Ža., since G1 ␣ 1 / G1 and G 2 ␣ 2 / G 2 . EXAMPLES 7.8. Internal direct products in which < G1 ␣ 2 < s 1 but < G 2 ␣ 1 < is any non-zero cardinal number at all, and the other traditions also can be violated.
623
INTERNAL DIRECT PRODUCTS
Let Ž I, . be any groupoid whose subgroupoid J has identity element j. ŽThis j need not be an identity for the whole of I.. Let and be mappings ⺞ = I ª ⺞ j 04 , arbitrary except that ᭙ n g ⺞, ᭙ y g J, ny s n y 1. Then G is the set Ž⺞ j 04. = I = J. ᭙ m, n g ⺞ j 04 , ᭙ u, ¨ g I, ᭙ x, y g J,
¡Ž Ž m y n. ¨ q n, u, xy . Ž m, u, x . Ž n, ¨ , y .  s~ Ž n, u¨ , xy . ¢Ž Ž n y m. u q m, ¨ , xy .
if m ) n if m s n if m - n
Ž m, u, x . ␣ 1 s Ž m q 1, u, j . Ž m, u, x . ␣ 2 s Ž 0, x, x . . After all that, it may be heartening to observe that Examples 7.5 are the special cases having I s J s j4 and nj s n y 1, with Ž n, j, j . abbreviated to n, and nj abbreviated to n . Even so, perhaps the check of 2.5Ža. ᎐ Žc. had better not be left to the hapless reader. First,
Ž m, u, x . ␣ 1 Ž n, ¨ , y . ␣ 2  s Ž m q 1, u, j . Ž 0, y, y .  s Ž Ž m q 1 . y , u, jy . s Ž m, u, y . Thus Ž a.
Ž b.
⬗ mq1)0 ⬗ y g J.
Ž m, u, x . ␣ 1 Ž m, u, x . ␣ 2  s Ž m, u, x . Ž m, u, x . ␣ 1 Ž n, ¨ , y . ␣ 2 ␣ 1 s Ž m, u, y . ␣ 1 s Ž m q 1, u, j . s Ž m, u, x . ␣ 1 Ž m, u, x . ␣ 1 Ž n, ¨ , y . ␣ 2 ␣ 2 s Ž m, u, y . ␣ 2 s Ž 0, y, y . s Ž n, ¨ , y . ␣ 2
Ž c . From the definitions of  and ␣ 1 ,
Ž m, u, x . Ž n, ¨ , y . ␣ 1
¡Ž Ž m y n. ¨ q n q 1, u, j . ¢Ž Ž n y m. u q m q 1, ¨ , j .
s~ Ž n q 1, u¨ , j .
if m ) n if m s n if m - n
s Ž m q 1, u, j . Ž n q 1, ¨ , j . 
⬗ jj s j
s Ž m, u, x . ␣ 1 Ž n, ¨ , y . ␣ 1  .
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K. E. PLEDGER
Ž m, u, x . Ž n, ¨ , y . ␣ 2 s Ž something, something, xy . ␣ 2 s Ž 0, xy , xy . s Ž 0, x, x . Ž 0, y, y .  s Ž m, u, x . ␣ 2 Ž n, ¨ , y . ␣ 2  . The definitions of ␣ 1 and ␣ 2 show that G1 s Ž p, u, j . : p g ⺞, u g I 4 , so G1 ␣ 2 s Ž0, j, j .4 ; and also that G 2 s Ž0, x, x . : x g J 4 , so G 2 ␣ 1 s Ž1, x, j . : x g J 4 . It follows easily that 2.6Žd. ᎐ Žf. are all false. However 2.6Ža. ᎐ Žc. depend upon the choice of I, J, , and . Conditions for 2.6Žc. are almost immediate. From the previous paragraph < G1 ␣ 2 < s 1, but the definition of  shows that Ž G 2 ␣ 1 ,  . ( Ž J, .. Thus G satisfies 2.6Žc. if and only if < J < s 1, i.e., J s j4 . However G 2 ␣ 1 can be isomorphic to any groupoid with identity, and hence of any non-zero order. This makes Examples 7.8 very different from orthodox internal direct products as in 2.6 or even in 3.4 where every member of G 2 has the same first component d. Next consider a condition which is sufficient Žand in fact necessary . for G to satisfy 2.6Žb.. Suppose ᭙ n g ⺞, ᭙ y g J, ny s n y 1. Then
Ž m q 1, u, j . Ž 0, y, y .  s Ž Ž m q 1 . y , u, jy . s Ž m, u, y . , and
Ž 0, y, y . Ž m q 1, u, j .  s Ž Ž m q 1 . y , u, yj . s Ž m, u, y . . Thus every element Ž m q 1, u, j . of G1 commutes with every element Ž0, y, y . of G 2 , so 2.6Žb. is satisfied. Now consider some conditions which are sufficient Žand in fact necessary. for G to satisfy 2.6Ža.. Suppose Ž I, . has an identity element i f J, and ᭙ n g ⺞, ni s ni s n. Then
Ž m, u, x . Ž 0, i , j .  s
½
Ž mi , u, xj . Ž 0, ui , xj .
if m ) 0 if m s 0
s Ž m, u, x . , and
Ž 0, i , j . Ž n, ¨ , y .  s
½
Ž 0, i¨ , jy . Ž ni , ¨ , jy .
s Ž n, ¨ , y . .
if 0 s n if 0 - n
INTERNAL DIRECT PRODUCTS
625
Thus G has the identity element Ž0, i, j ., so 2.6Ža. is satisfied. Notice that G1 l G 2 is empty, yet G has an identity element somewhere else. All those conditions suggest examples to separate some of the classes in Fig. 4.5. First:
Ž I, .
i
j
i j
i j
j j
with J s j 4
and ni s ni s n,
᭙ n g ⺞,
nj s n y 1.
The conditions used in defining G are fulfilled, and the foregoing discussion shows that G satisfies 2.6Ža. and Žc.. It also satisfies 2.6Žb. if ᭙ n g ⺞, nj s n y 1, which provides an example satisfying 2.6Ža᎐c. but neither Žd., Že., nor Žf.. But otherwise ᭚ n g ⺞: nj / n y 1, say, for example, 5 j s 8. In that case Ž5, i, j .Ž0, j, j .  s Ž4, i, j . / Ž8, i, j . s Ž0, j, j .Ž5, i, j .  contrary to 2.6Žb.; which provides an example satisfying 2.6Ža, c. but neither Žb., Žd., Že., nor Žf.. Next:
Ž I, .
i
j
k
i j k
i j k
j j k
k k k
with J s j, k 4 ,
and ᭙ n g ⺞,
ni s ni s n,
nj s nk s nj s nk s n y 1.
The conditions used in defining G are fulfilled, and the earlier discussion shows that G satisfies 2.6Ža, b. but neither Žc., Žd., Že., nor Žf.. Finally:
Ž I, .
j
k
j k
j k
k k
with J s I,
and ᭙ n g ⺞,
nj s nk s n y 1,
but 5 j s 8. The reader may check that G is then an internal direct product satisfying none of the six conditions 2.6Ža. ᎐ Žf.. This highly unorthodox example vindicates the claim at the end of Section 2.
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K. E. PLEDGER
EXAMPLES 7.9. Internal direct products in which < G 2 ␣ 1 < and < G1 ␣ 2 < are any non-zero cardinal numbers. The construction details will not be given. But just as each Example 7.7 is compounded from two versions of 7.5, so each Example 7.9 can be constructed analogously from two versions of 7.8. The elements are now ordered sextuples so the notation looks intricate, but the generalization from Examples 7.8 raises no other difficulties.
8. EPILOGUE Examples 7.5᎐7.9 perhaps look like conjuring tricks. It seems only fair to disclose briefly how the rabbits got into the hats. Consider any internal direct product Ž G,  , ␣ 1 , ␣ 2 .. For all non-negative integers m and n, define ␥m n : G ª G by x␥m n s x ␣ 1mq 1 x ␣ 2nq 1 . Every such ␥m n is an endomorphism of G Žas shown in Theorem 2.5Ža. for ␥ 00 and in Lemma 6.2 for ␥ 11 .. If Gm n is defined as G␥m n , then Gm n l Gp q s G Ma xŽ m, p., MaxŽ n, q. . This can be extended by defining G⬁0 s F Gm 0 : m g ⺞4 and G 0⬁ s F G 0 n : n g ⺞4 and then ᭙ r, s g 04 j ⺞ j ⬁4 , Gr s s Gr 0 l G 0 s . Incidentally the Theorem 3.3 conditions are also equivalent to G10 s G, the same conditions with transposed subscripts are equivalent to G 01 s G, and the Theorem 3.4 conditions are equivalent to G10 s G s G 01 . G can now be partitioned into four subsets as follows. Define K s G y Ž G10 j G 01 ., L1 s G⬁0 y G 01 , and L2 s G 0⬁ y G10 . Also write ⌫ s ␥m n : m, n g ⺞ j 044 , so, for example, K ⌫ s x␥m n : x g K, m G 0, n G 04 . Then G is the disjoint union K ⌫ j ˙ L1 ⌫ j ˙ L2 ⌫ j ˙ G⬁⬁. The Theorem 3.4 conditions such as < G1 l G 2 < s 1 are equivalent to K s L1 s L2 s ⭋, i.e., G s G⬁⬁ . ŽLemma 6.5 partly proves this.. Examples 7.5 and 7.8 have K s L1 s G⬁⬁ s ⭋, so G s L2 ⌫. Examples 7.5 were constructed by writing L2 s 04 and each 0␥m n s m Žand in fact include isomorphic copies of all groupoids L2 ⌫ having < L2 < s 1.. After a long struggle, Examples 7.8 were constructed by writing L2 s 04 = I = J and each Ž0, u, x .␥m n s Ž m, u, x .. Examples 7.7 and 7.9 have L1 s L2 s G⬁⬁ s ⭋, so G s K ⌫. Examples 7.7 were constructed by writing K s Ž0, 0.4 and each Ž0, 0.␥m n s Ž m, n.. Examples 7.9 have K s 04 = I1 = J1 = 04 = I2 = J 2 . An obvious final question Žvital for the refinement property of w5x etc.. is how to generalize this whole theory of internal direct products to more than the two factors G1 and G 2 . The ideal would be to find minimum restrictions permitting such a generalization. Some restrictions would probably be needed because Examples 7.5 display a non-associative aspect of internal direct multiplication, and perhaps also because the related
INTERNAL DIRECT PRODUCTS
627
Example 7.6 Žwith k s 0. has one factorization which satisfies the conditions 2.6Žc᎐e. and another which does not.
REFERENCES 1. P. M. Cohn, ‘‘Universal Algebra,’’ Harper & Row, 1965; Reidel, 1981. 2. P. Crawley and B. Jonsson, Refinements for infinite direct decompositions, Pacific J. Math. ´ 14 Ž1964., 797᎐855. 3. J. M. G. Fell and A. Tarski, On algebras whose factor algebras are Boolean, Pacific J. Math. 2 Ž1952., 297᎐318. 4. B. Jonsson, The unique factorization problem for finite relational structures, Colloq. Math. ´ 14 Ž1966., 1᎐32. 5. B. Jonsson and A. Tarski, ‘‘Direct Decompositions of Finite Algebraic Systems,’’ Notre ´ Dame Mathematical Lectures No. 5, Notre Dame, Indiana, 1947. 6. J. Łos, ´ Direct sums in general algebra, Colloq. Math. 14 Ž1966., 33᎐38. 7. R. Willard, Varieties having Boolean factor congruences, J. Algebra 132 Ž1990., 130᎐153.
Internal Direct Products of Groupoids K. E. Pledger Mathematics Department, Victoria Uni¨ ersity of Wellington, P.O. Box 600, Wellington, New Zealand Communicated by Leonard Lipshitz Received June 30, 1998
When generalizing the internal direct product from groups to all groupoids Žbinary systems., it has been customary to imitate the group case by making restrictions, especially the existence of an identity element. This article develops what seems a natural basic definition of internal direct product, then uses it as a background against which to compare more popular restricted versions. 䊚 1999 Academic Press
1. PROLOGUE The fruitfulness of Jonsson and Tarski’s work w5x on direct product ´ decompositions is very clear from Willard’s summary w7, pp. 130᎐131x. Yet w5x includes a curious oversight concerning the definition of internal direct products of groupoids. That definition generalizes the familiar group case, which can be described as follows without mentioning normal subgroups. A group G may be the internal direct product of subgroups G1 and G 2 . In that case each element of G is uniquely expressible as x 1 x 2 with x 1 g G1 , x 2 g G 2 ; and the product of any such x 1 x 2 and y 1 y 2 can be written Ž x 1 y 1 .Ž x 2 y 2 .. The connection with the ordinary external direct product G1 = G 2 is that the mapping : Ž x 1 , x 2 . ¬ x 1 x 2 is an isomorphism from G1 = G 2 onto G. If G is an external direct product H1 = H2 of groups, then it is also an internal direct product as follows. Let e1 be the identity element of H1 , and e2 of H2 , then let G1 s H1 = e2 4 and G 2 s e14 = H2 . It is easy to 599 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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K. E. PLEDGER
check that G is the internal direct product of G1 and G 2 , although the isomorphism is the notationally clumsy
Ž Ž x 1 , e 2 . , Ž e1 , x 2 . . ¬ Ž x 1 , e 2 . Ž e1 , x 2 . s Ž x 1 , x 2 . . Such arguments depend so vitally upon e1 and e2 that past generalizations of the internal direct product from groups to groupoids have treated identity elements as indispensable. The classic statement of that belief occurs within the book w5x of Jonsson and Tarski. Their algebras are ´ groupoids with multiple operators, but the essential problem with internal direct products lies in the groupoid structure. On p. 7 they make the following claim. ‘‘On the other hand, the definition of a winternal x direct product implicitly involves the notion of a zero element, and it cannot be applied to systems without a zero unless we agree to use the term ‘‘subalgebra’’ in a more general sense. What is even more important, a detailed examination of the proof of 1.5 wcharacterizing internal direct productsx reveals that condition 1.1Žii. wexistence and properties of zerox plays an essential role in this proof. It can easily be shown by means of examples that, in general, Theorem 1.5 does not apply to systems A in which either part Žii⬘. or part Žii⬙ . of this condition fails; for it may then happen that A is isomorphic to the cardinal wi.e., external directx product of two systems B and C, without containing any subsystem isomorphic to B or C. We can thus say that only by restricting ourselves to systems which satisfy 1.1Žii. are we in a position to introduce an adequate notion of an inner direct product.’’ Example 7.1 illustrates the difficulty they described and firmly blamed on the lack of an identity element. The strong but unproved statement of Jonsson and Tarski seems to have started a tradition, and not only in later ´ papers involving the same authors w2᎐4x. Łos´ w6x uses more general algebras, but imposes conditions from which he can derive essentially the existence of the groupoid identity element as a theorem. That tradition is renounced by Definition 2.2 below, whose consequences do not include the existence of an identity element. It seems the most natural and economical definition of internal direct product. ŽEvidently Cohn thinks so, as his w1, p. 56, Exercise 11x gives the same definition; but he then reverts to tradition by citing w5x..
2. FUNDAMENTALS The notation is reversed Polish. Various groupoids use the same operator symbol  , so the result of the appropriate binary operation on x and y is written xy  . Also the notation introduced in each definition is assumed from then on.
601
INTERNAL DIRECT PRODUCTS
DEFINITION 2.1. The external direct product of groupoids Ž G1 ,  . and Ž G 2 ,  . is Ž G1 = G 2 ,  ., where ᭙ x 1 , y 1 g G1 ,
᭙ x 2 , y 2 g G2 ,
Ž x1 , x 2 . Ž y1 , y 2 .  s Ž x1 y1  , x 2 y 2  . . DEFINITION 2.2. A groupoid Ž G,  . is the internal direct product of its non-empty subgroupoids G1 and G 2 if and only if the mapping : Ž x 1 , x 2 . ¬ x 1 x 2  is an isomorphism from G1 = G 2 onto G. Prescribing that G / ⭋ is a minor convenience. Of course is just the mapping which interprets the binary operator symbol  in G. Next, the isomorphism y1 can be used to define the projections ␣ 1 and ␣ 2 from G onto G1 and G 2 . DEFINITION 2.3. With the notation of Definition 2.2, ᭙ x g G, x y1 s Ž x ␣ 1 , x ␣ 2 . . From Definitions 2.2 and 2.3 easily follow G␣ 1 s G1 and G␣ 2 s G 2 . Hence an element of G1 may be written either x 1 or x ␣ 1 as convenient, and a similar remark applies to G 2 . This is assumed in, for example, the proof of Theorem 2.5Žb.. Also the consequence that G1 ␣ 1 : G1 and G 2 ␣ 2 : G 2 is useful from Theorem 3.2 onward. THEOREM 2.4. ᭙ x 1 , y 1 g G 1 , ᭙ x 2 , y 2 g G 2 , x 1 x 2  y 1 y 2  s x 1 y 1  x 2 y 2  ; i.e., elements can be combined componentwise. Proof. x 1 x 2  y 1 y 2  s Ž x 1 , x 2 . Ž y 1 , y 2 .  s Ž x 1 , x 2 . Ž y 1 , y 2 . 
2.2
s Ž x1 y1  , x 2 y 2  .
2.1
s x 1 y 1  x 2 y 2  .
2.2
The second paragraph of Section 1 has now been formalized for general groupoids. Such pedestrian results are easily obtained by going back and forth to the external direct product, but for faster progress an internal direct product can be characterized purely by intrinsic properties. THEOREM 2.5. Suppose a set G is closed to a binary operation  and unary operations ␣ 1 and ␣ 2 . The groupoid Ž G,  . is the internal direct product of G␣ 1 and G␣ 2 if and only if the algebra Ž G,  , ␣ 1 , ␣ 2 . has the
602
K. E. PLEDGER
following properties: Ža. ᭙ x g G, x ␣ 1 x ␣ 2  s x Žb. ᭙ x, y g G, x ␣ 1 y␣ 2 ␣ 1 s x ␣ 1 and x ␣ 1 y␣ 2 ␣ 2 s y␣ 2 Žc. ᭙ x, y g G, x ␣ 1 y␣ 1  s xy ␣ 1 and x ␣ 2 y␣ 2  s xy ␣ 2 . Proof. Write G␣ 1 s G1 and G␣ 2 s G 2 . First, assume Ž G,  . is the internal direct product of G1 and G 2 . Then:
Ž a.
x s x y1 s Ž x ␣ 1 , x ␣ 2 .
᭙ xgG,
2.3
s x ␣1 x ␣2 
Ž b.
᭙ x 1 g G1 ,
2.2
᭙ y 2 g G2 ,
Ž x 1 , y 2 . s Ž x 1 , y 2 . y1 s x 1 y 2 y1 s Ž x 1 y 2 ␣ 1 , x 1 y 2 ␣ 2 .
2.2 2.3
so x 1 s x 1 y 2 ␣ 1 and y 2 s x 1 y 2 ␣ 2
Ž c.
᭙ x, ygG,
Ž xy ␣ 1 , xy ␣ 2 . s xy y1 y1
s x
2.3 y1
⬗ y1 is an isomorphism by 2.2
y 
s Ž x ␣ 1 , x ␣ 2 . Ž y␣ 1 , y␣ 2 . 
2.3
s Ž x ␣ 1 y␣ 1  , x ␣ 2 y␣ 2  .
2.1
so xy ␣ 1 s x ␣ 1 y␣ 1  and xy ␣ 2 s x ␣ 2 y␣ 2  . Conversely, assume Ža. ᎐ Žc. are satisfied. Define : G ª G1 = G 2 by x s Ž x ␣ 1 , x ␣ 2 .. Then x s y
«
x ␣ 1 s y␣ 1
«
x ␣ 1 x ␣ 2  s y␣ 1 y␣ 2 
«
xsy
and
x ␣ 2 s y␣ 2
by Ž a . . ⬖ is injective.
Also ᭙ x 1gG1 ,
᭙ y 2 g G2 , s Ž x1 , y2 .
x 1 y 2  s Ž x 1 y 2 ␣ 1 , x 1 y 2 ␣ 2 . by Ž b . . ⬖ is surjective.
INTERNAL DIRECT PRODUCTS
603
Also ᭙ x, ygG,
x y s Ž x ␣ 1 , x ␣ 2 . Ž y␣ 1 , y␣ 2 .  s Ž x ␣ 1 y␣ 1  , x ␣ 2 y␣ 2  .
2.1
s Ž xy ␣ 1 , xy ␣ 2 .
by Ž c .
s xy  .
⬖ is a homomorphism.
Finally let s y1 . Then G is an internal direct product as in Definition 2.2. Theorem 2.5 offers a useful alternative point of view. The class of all groupoid internal direct products may now be regarded as a variety of Ž  , ␣ 1 , ␣ 2 .-algebras, with the formulae 2.5Ža. ᎐ Žc. as axioms. ŽThese axioms are in fact independent.. Subsequent proofs refer back to 2.5Ža. ᎐ Žc. rather than directly to Definition 2.2. Between those general proofs there are also references to special properties commonly expected of internal direct products. The following list seems to cover most such properties. CONDITIONS 2.6. Custom prefers an internal direct product to satisfy: Ža. G has a neutral Židentity. element. Žb. Every element of G1 commutes with every element of G 2 . Žc. < G 2 ␣ 1 < s < G1 ␣ 2 < s 1. Žd. G 2 ␣ 1 s G1 ␣ 2 s a singleton invariant under ␣ 1 and ␣ 2 . Že. < G1 l G 2 < s 1. Žf. ␣ 1 < G 1 and ␣ 2 < G 2 are identity mappings. Equivalently, ␣ 12 s ␣ 1 and ␣ 22 s ␣ 2 . The statement of Žd. may appear awkward, especially since the singleton in Žd. and Že. traditionally contains just the identity element mentioned in Ža.. However the six statements Ža. ᎐ Žf. as they stand are useful for reference later. Rather obviously, they are not independent. Much less obviously, they can all be false. ŽCf. the last of Examples 7.8..
3. BASIC THEOREMS Examples 7.3᎐7.9 show that 2.6Žf. cannot be proven in general. However 2.5Žc. means that ␣ 1 and ␣ 2 must be endomorphisms, and Theorem 3.2 shows that their restrictions to G1 and G 2 , respectively, must be injective endomorphisms.
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K. E. PLEDGER
First it is convenient to have notation for right and left translations. DEFINITION 3.1. For any a g G, ᭙ x g G, x a s xa  and x a s ax  . THEOREM 3.2. In any internal direct product Ž G,  , ␣ 1 , ␣ 2 . with G1 s G␣ 1 and G 2 s G␣ 2 , ␣ 1 < G 1 and ␣ 2 < G 2 are injections. Proof. Choose any a2 g G 2 Žwhich is non-empty by 2.2.. Then ᭙ x 1gG1 ,
x 1 ␣ 1 a 2 ␣ 1 s x 1 ␣ 1 a2 ␣ 1 
3.1
s x 1 a2 ␣ 1
2.5 Ž c .
s x1 .
2.5 Ž b .
Thus ␣ 1 < G 1 has a right inverse, so it is injective. Similarly ␣ 2 < G 2 is injective. COROLLARY. For any a2 g G 2 , a 2 ␣ 1 < G 1␣ 1 is a right in¨ erse of ␣ 1 < G 1. For any a1 g G1 , a1␣ 2 < G 2 ␣ 2 is a right in¨ erse of ␣ 2 < G 2 . In light of Theorems 2.5Žc. and 3.2, the question whether ␣ 1 < G 1 is an actual automorphism is simply the question whether G1 ␣ 1 s G1. Examples 7.5᎐7.9 show that this is not always so. But the condition G1 ␣ 1 s G1 is a significant restriction Žas is G 2 ␣ 2 s G 2 ., which the next theorem expresses in various forms. THEOREM 3.3. In any internal direct product Ž G,  , ␣ 1 , ␣ 2 . with G1 s G␣ 1 and G 2 s G␣ 2 , the following conditions are equi¨ alent. Ža. G1 ␣ 1 s G1. Žb. G 2 ␣ 1 is a singleton d14 with d1 ␣ 1 s d1. Žc. G1 ␣ 1 l G 2 ␣ 1 is a singleton d14 . Žd. G1 ␣ 1 l G 2 ␣ 1 / ⭋. Four other mutually equi¨ alent conditions are obtained by transposing the subscripts 1 and 2 abo¨ e. Proof. Ža. « Žb.. Choose any c1 g G1 Žwhich is non-empty.. Then c1 ␣ 2 ␣ 1 g G1 s G1 ␣ 1 from Ža., so ᭚d1 g G1: c1 ␣ 2 ␣ 1 s d1 ␣ 1. ⬖ ᭙ x 2gG 2 ,
x 2 ␣ 1 s c1 x 2 ␣ 2 ␣ 1 s c1 ␣ 2 ␣ 1 x 2 ␣ 2 ␣ 1 
2.5 Ž b . 2.5 Ž c .
s d1 ␣ 1 x 2 ␣ 2 ␣ 1  s d 1 x 2 ␣ 2 ␣ 1
2.5 Ž c .
s d1 ,
2.5 Ž b .
605
INTERNAL DIRECT PRODUCTS
⬖G 2 ␣ 1 s d14 , so d1 is unique. Also d1 ␣ 1 s c1 ␣ 2 ␣ 1 g G 2 ␣ 1 s d14 . Hence Žb.. Žb. « Žc.. From Žb., G 2 ␣ 1 s d1 4 s d1 ␣ 1 4 : G1 ␣ 1. Hence Žc.. Žc. « Žd. obviously. Žd. « Ža.. From Žd., ᭚a1 g G1 , ᭚a2 g G 2 : a1 ␣ 1 s a2 ␣ 1. ⬖ ᭙ x 1gG1 ,
x 1 s x 1 a2 ␣ 1 s x 1 ␣ 1a 2 ␣ 1 
2.5 Ž b . 2.5 Ž c .
s x 1 ␣ 1a 1 ␣ 1  s x 1 a 1 ␣ 1
2.5 Ž c .
g G1 ␣ 1 . ⬖G1 : G1 ␣ 1. Also G1 ␣ 1 : G␣ 1 s G1. Hence Ža.. Similar proofs apply throughout when subscripts 1 and 2 are transposed. COROLLARY. Under the conditions 3.3Ža. ᎐ Žd., d1 < G 1 is the Ž two-sided. in¨ erse automorphism of ␣ 1 < G 1. Under the corresponding conditions with transposed subscripts, d 2 < G 2 is the Ž two-sided. in¨ erse automorphism of ␣2 < G2. Proof. Rewrite the corollary to Theorem 3.2 using 3.3Ža. and Žb., and also 2.5Žc.. Of the two statements in 3.3Žb., the invariance under ␣ 1 is the vital one. Condition 3.3Žb. can in fact be weakened to G 2 ␣ 12 s G 2 ␣ 1 with impunity, but Examples 7.5᎐7.7 show that it cannot be weakened merely to < G 2 ␣ 1 < s 1. Conditions 3.3Ža. ᎐ Žd. are not equivalent to the four conditions obtained from them by transposing subscripts. Examples 7.6 Žwith k G 1. satisfy the former but not the latter, and Examples 7.5 and 7.8 satisfy the latter but not the former. However the next theorem deals with the situation where both are satisfied simultaneously. THEOREM 3.4. In any internal direct product Ž G,  , ␣ 1 , ␣ 2 . with G1 s G␣ 1 and G 2 s G␣ 2 , the following conditions are equi¨ alent. Ža. G1 ␣ 1 s G1 and G 2 ␣ 2 s G 2 . Žb. G 2 ␣ 1 s G1 ␣ 2 s a singleton d4 with d ␣ 1 s d ␣ 2 s d. Žc. G1 l G 2 s a singleton d4 . Žd. G1 l G 2 / ⭋. Proof. Ža. « Žb.. From Ža., Theorem 3.3 gives G 2 ␣ 1 s d1 4 with d1 ␣ 1 s d1 , and G1 ␣ 2 s d 2 4 with d 2 ␣ 2 s d 2 . Also d 2 ␣ 1 g G 2 ␣ 1 s
606
K. E. PLEDGER
d14 , so d 2 ␣ 1 s d1. Similarly d1 ␣ 2 s d 2 . ⬖ d1 s d 1 ␣ 1 d 1 ␣ 2 
2.5 Ž a .
s d1 d 2  s d 2 ␣ 1 d 2 ␣ 2  s d 2
2.5 Ž a .
s d, say. Hence Žb.. Žb. « Žc.. From Žb., d g G 2 ␣ 1 l G1 ␣ 2 : G1 l G 2 . Also ᭙ x g G1 l G 2 , x ␣ 1 g Ž G1 l G 2 . ␣ 1 : G 2 ␣ 1 s d4 and x ␣ 2 g Ž G1 l G 2 . ␣ 2 : G1 ␣ 2 s d 4 , so x s x ␣1 x ␣2 
2.5 Ž a .
s dd  s d
⬗ d 4 is a subgroupoid.
Hence Žc.. Žc. « Žd. obviously. Žd. « Ža.. Ž G1 l G 2 . ␣ 1 : G1 ␣ 1 l G 2 ␣ 1. Hence G1 l G 2 / ⭋
«
G1 ␣ 1 l G 2 ␣ 1 / ⭋
«
G1 ␣ 1 s G1
3.3
and similarly G2 ␣ 2 s G2 .
COROLLARY 1. Under the conditions 3.4Ža. ᎐ Žd., d < G 1 and ␣ 1 < G 1 are in¨ erse automorphisms, d < G 2 and ␣ 2 < G 2 are in¨ erse automorphisms, and dd  s d. Proof. In the corollary to Theorem 3.3, put d1 s d 2 s d. Also observe that d4 is a subgroupoid. COROLLARY 2.
< G1 l G 2 < can only be 1 or 0.
Proof. 3.4Žc. m 3.4Žd.. Cases with < G1 l G 2 < s 1 satisfy all the Theorem 3.4 conditions and clearly generalize traditional internal direct products, whereas cases with < G1 l G 2 < s 0 satisfy none of the Theorem 3.4 conditions and are much more novel. Examples 7.5᎐7.9 Žand Section 8. exhibit some of the tangled consequences of G1 l G 2 s ⭋. In the meantime Theorems 3.5᎐6.6 explore the tidier situation where G1 l G 2 s d4 .
INTERNAL DIRECT PRODUCTS
607
First, Theorem 3.4 Corollary 1 has something like a converse. THEOREM 3.5. If a groupoid Ž G1 ,  . has an idempotent element d1 such that d1 is an automorphism, then G1 is isomorphic to the left factor of an internal direct product. If a groupoid Ž G 2 ,  . has an idempotent element d 2 such that d 2 is an automorphism, then G 2 is isomorphic to the right factor of an internal direct product. Any such G1 can be combined with any such G 2 , and the resulting internal direct product satisfies the Theorem 3.4 conditions. Proof. Both G1 and G 2 will be needed in the construction. So, if given only G1 , let G 2 be the trivial group d 2 4 ; if given only G 2 , let G1 be the trivial group d14 . Then in all cases both G1 and G 2 are available. d1 is an automorphism of G1 , so d1y1 is also. Likewise d 2y1 is an automorphism of G 2 . Hence we can define within G1 = G 2
Ž x 1 , x 2 . ␣ 1 s ž x 1 d1y1 , d 2 / , and
Ž x 1 , x 2 . ␣ 2 s ž d1 , x 2 d 2y1 / . The verification of 2.5Ža. ᎐ Žc. is then quite routine, and is omitted here. Also
Ž G1 = G 2 . ␣ 1 s G1 = d 2 4 ( G1
⬗ d1y1 is an automorphism of G1 ⬗ d 2 is idempotent,
and similarly Ž G1 = G 2 . ␣ 2 s d14 = G 2 ( G 2 . Thus Ž G1 = G 2 ,  . is an internal direct product of subgroupoids isomorphic to G1 and G 2 . These subgroupoids have intersection Ž G1 = d 2 4. l Ž d14 = G 2 . s Ž d1 , d 2 .4 as in Theorem 3.4Žc.. The main difficulty in applying Theorem 3.5 is notational. In practice the construction is easier if the embeddings of G1 and G 2 are treated as identifications. For this write d1 s d 2 s d, choose notation to make G1 l G 2 s d4 only, and write the elements of G as x 1 x 2  rather than Ž x 1 , x 2 .. Then G1 and G 2 are themselves subgroupoids of G, as in Examples 7.2᎐7.4. The idempotence condition dd  s d is vital to the Theorem 3.5 construction. In Example 7.1, both a and b are automorphisms of G1 , and both f and g are automorphisms of G 2 ; but the direct product cannot be made internal as above, because none of a, b, f, and g is idempotent.
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K. E. PLEDGER
4. ANALYZING THE TRADITION It is now possible to look in more detail at the traditional properties 2.6Ža. ᎐ Žf. and relations between them. Theorem 3.4 already includes 2.6Žd. and Že., and the following theorems treat internal direct products which are closer to tradition in various other ways. The first of them involves 2.6Že. and Žb.. THEOREM 4.1. If G1 l G 2 s d4 , and d commutes with e¨ ery element of G, then e¨ ery element of G1 commutes with e¨ ery element of G 2 . Proof. ᭙ x 1 g G1 , ᭙ x 2 g G 2 , x 2 x 1  s dx 2 ␣ 2  x 1 ␣ 1 d 
2.5 Ž a . , 3.4 Corollary 1
s dx 1 ␣ 1  x 2 ␣ 2 d 
2.4
s x 1 ␣ 1 d  dx 2 ␣ 2 
if d commutes with every element
s x1 x 2  .
2.5 Ž a . , 3.4 Corollary 1
Ž ⬗ d g G1 and d g G2 .
This generalizes the traditional proof which assumes an identity element and simply substitutes it for x 1 and y 2 in Theorem 2.4. The converse of Theorem 4.1 is false. Example 7.6 Žwith k s 1. has every element of G1 commuting with every element of G 2 , but G1 l G 2 empty. Now, 2.6Žf. includes the claim that ␣ 1 < G 1 is an identity mapping. Theorem 3.3 and its corollary give conditions merely for ␣ 1 < G 1 to be an automorphism, but by an easy deduction it is idempotent and hence the identity mapping of G1 if and only if G 2 ␣ 1 s d14 where d1 is a right identity element for G1. Rather than spend time on that and on the corresponding result for ␣ 2 , the next theorem strengthens Theorem 3.4 Žinvolving both ␣ 1 and ␣ 2 . to the whole of 2.6Žf.. THEOREM 4.2. ␣ 1 < G 1 and ␣ 2 < G 2 are both identity mappings Ž i.e., ␣ 12 s ␣ 1 and ␣ 22 s ␣ 2 . if and only if some element d in G1 l G 2 is both a right identity element for G1 and a left identity element for G 2 . In such a case d4 s G 1 l G 2 s G 2 ␣ 1 s G 1 ␣ 2 . Proof. Assume both ␣ 1 < G 1 s 1 G 1 and ␣ 2 < G 2 s 1 G 2 . This is a special case of 3.4Ža.. Hence 3.4Žc. and Žb. give d4 s G1 l G 2 s G 2 ␣ 1 s G1 ␣ 2 , and 3.4 Corollary 1 gives d < G 1 s 1 G 1 and d < G 2 s 1 G 2 so d has the required identity properties.
INTERNAL DIRECT PRODUCTS
609
Conversely, assume an element d in G1 l G 2 is both a right identity element for G1 and a left identity element for G 2 . Then 3.4Žd. holds, so 3.4Žc. and Žb. give d4 s G1 l G 2 s G 2 ␣ 1 s G1 ␣ 2 , and 3.4 Corollary 1 gives ␣ 1 < G 1 s 1 G 1 and ␣ 2 < G 2 s 1 G 2 . Łos´ included 2.6Žf. in his definition of internal direct product w6, p. 34, Ž1.x, but he missed the consequence that G1 l G 2 / ⭋. ŽCf. w6, p. 35, Lemma 3x.. An effect of combining 2.6Žb. and Žf. can be inferred from 4.2 as THEOREM 4.3. If e¨ ery element of G1 commutes with e¨ ery element of G 2 , and ␣ 12 s ␣ 1 and ␣ 22 s ␣ 2 , then G has an identity element. Proof. Assuming ␣ 12 s ␣ 1 and ␣ 22 s ␣ 2 , apply Theorem 4.2 but write e instead of d. Then in particular e g G1 , and e is a left identity element for G 2 . Thus, assuming every element of G1 commutes with every element of G 2 , e is also a right identity element for G 2 . ⬖ ᭙ xgG,
xe  sx ␣ 1 x ␣ 2  ee 
2.5 Ž a .
s x ␣ 1 e  x ␣ 2 e 
2.4, ⬗ e g G1 and e g G 2
s x ␣1 x ␣2 
⬗ e is a right identity for G1 and G 2
sx
2.5 Ž a .
s ex 
similarly.
The last of these miscellaneous theorems combines 2.6Ža. and Že.. THEOREM 4.4. d s e.
If G has identity element e, and G1 l G 2 s d4 , then
Proof. Assume G1 l G 2 s d4 as in 3.4Žc.. Then 3.4Žb. gives d s d ␣ 1 s d ␣ 2 , so d ␣ 1 s d ␣ 12 s ed ␣ 12 s e ␣ 1 d ␣ 1 ␣ 1
assuming identity element e 2.5 Ž c .
s e ␣ 1 d ␣ 2 ␣ 1 s e␣1 ,
2.5 Ž b .
and similarly d ␣ 2 s e ␣ 2 . Combine these two equations using 2.5Ža..
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K. E. PLEDGER
Theorem 4.4 does not assert that if G has identity element e, then G1 l G 2 s e4 : the hypothesis G1 l G 2 s d4 is vital. Some of Examples 7.8 show that when G1 l G 2 is empty, G may yet have an identity element somewhere else. Implications connecting 2.6Ža. ᎐ Žf. can now be assembled. Theorem 4.2 shows that Žf. « Že.. Theorem 3.4 shows that Že. m Žd.. Trivially Žd. « Žc.. Theorems 4.4 and 4.1 show that Ža. n Že. « Žb.. Theorems 4.4 and 4.2 show that Ža. n Že. « Žf.. Theorem 4.3 shows that Žb. n Žf. « Ža.. From all that, the appropriate Boolean disentangling sorts internal direct products into 12 classes partially ordered by inclusion as in Fig. 4.5. ŽFor simplicity, one-sided conditions such as in Theorem 3.3 are not included here.. The figure’s notation Žc, d, e., for example, means the class of all internal direct products satisfying 2.6Žc., Žd., and Že.. The 12 classes in the figure are all distinct. Example 7.2 satisfies 2.6Žc᎐f. but neither Ža. nor Žb., so every class containing Žc, d, e, f. is distinct from every class contained in Ža. or Žb.. Example 7.3 shows likewise that every class containing Žb, c, d, e. is distinct from every class contained in Ža. or Žc, d, e, f.. Various cases of Example 7.8 show that every class containing Ža, b, c. is distinct from every class contained in Žc, d, e., every class containing Ža, b. is distinct from every class contained in Žc., and every class containing Ža, c. is distinct from every class contained in Žb.. Perhaps Theorem 3.4 Corollary 2 locates the major watershed, between internal direct products having < G1 l G 2 < s 1 and those having < G1 l G 2 < s 0, i.e., between members and non-members of Žc, d, e.. Within that class, Fig. 4.5 shows that Žb. and Žf. Žessentially as in Theorems 4.1 and 4.2. may be regarded as two independent components which together strengthen the condition < G1 l G 2 < s 1 to the fully classical Ža, b, c, d, e, f..
Ž All internal direct products. Ža.
Žb.
Žc.
Ža, c. Ža, b.
Žc, d, e. Žb, c.
Ža, b, c.
Žb, c, d, e. Ža, b, c, d, e, f.
FIGURE 4.5
Žc, d, e, f.
INTERNAL DIRECT PRODUCTS
611
5. STANDARD SPECIAL PROPERTIES Some familiar groupoid properties force internal direct products to satisfy 2.6Žc, d, e. or even Žc, d, e, f.. The next three theorems show why. As usual G / ⭋ from Definition 2.2. THEOREM 5.1.
If G is finite, then < G1 l G 2 < s 1.
Proof. By Theorem 3.2, ␣ 1 < G 1 and ␣ 2 < G 2 are injective. Hence in the finite case G1 ␣ 1 s G1 and G 2 ␣ 2 s G 2 . The result now follows from Theorem 3.4. THEOREM 5.2. < G1 l G 2 < s 1.
If G satisfies either the right or left cancellation law, then
Proof. Assume G satisfies the left cancellation law. Choose any a1 g G1 and a2 g G 2 . Then a1 ␣ 1 a1 ␣ 2  s a1
2.5 Ž a .
s a1 a2 ␣ 1
2.5 Ž b .
s a1 ␣ 1 a2 ␣ 1  .
2.5 Ž c .
Cancel a1 ␣ 1 from the left. ⬖ a1 ␣ 2 s a2 ␣ 1. ⬖ G1 l G 2 / ⭋. The result now follows from Theorem 3.4. The proof is similar if G has right cancellation. It may be observed that the foregoing proof really requires only one left cancellable element in G1 ␣ 1 or one right cancellable element in G 2 ␣ 2 . The last two theorems show that either finiteness or a cancellation law puts an internal direct product into the class Žc, d, e. of Fig. 4.5. However, it need not be in either of the subclasses Žb, c, d, e. or Žc, d, e, f.: Example 7.4 is a finite cancellation groupoid in which d is neither central Žas in 4.1. nor a one-sided identity element for G1 or G 2 Žas in 4.2.. The next result is stronger: the associative law puts an internal direct product into the class Žc, d, e, f.. In fact 2.6Žf. is equivalent to a restricted associative condition, as follows. THEOREM 5.3. ᭙ x 1 g G1 ,
␣ 12 s ␣ 1 and ␣ 22 s ␣ 2 if and only if ᭙ y g G,
᭙ z 2 g G2 ,
x 1 y  z 2  s x 1 yz 2  .
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K. E. PLEDGER
Proof. Assume every x 1 y  z 2  s x 1 yz 2  . Choose any a2 , b 2 g G 2 . Then ᭙ x 1gG1 ,
x 1 ␣ 1 s x 1 ␣ 1 a2 b 2 ␣ 1
2.5 Ž b .
s x 1 ␣ 1 a2  b 2 ␣ 1
from the above assumption
s x 1 ␣ 1 a2 ␣ 1 b 2 ␣ 1 
2.5 Ž c .
s x1 ␣ 1 b2 ␣ 1 
2.5 Ž b .
s x 1 b 2 ␣ 1
2.5 Ž c .
s x1 .
2.5 Ž b .
⬖ ␣ 1 < G 1 s 1 G 1. Similarly ␣ 2 < G 2 s 1 G 2 . Conversely, assume ␣ 12 s ␣ 1 and ␣ 22 s ␣ 2 . Then an element d has properties as in Theorem 4.2. Now, for every x 1 g G1 and y g G, x 1 y ␣ 1 s x 1 ␣ 1 y␣ 1  ⬖ x 1 y ␣ 1 d  s x 1 ␣ 1 y␣ 1 d 
2.5 Ž c . ⬗ d is a right identity for G1 , by 4.2.
Hence, for every z 2 g G 2 , x 1 y ␣ 1 z 2 ␣ 1  s x 1 ␣ 1 y␣ 1 z 2 ␣ 1 
⬗ G 2 ␣ 1 s d 4 by 4.2
⬖ x 1 y  z 2 ␣ 1 s x 1 ␣ 1 yz 2 ␣ 1 
2.5 Ž c .
s x 1 yz 2 ␣ 1 .
2.5 Ž c .
Similarly x 1 y  z 2 ␣ 2 s x 1 yz 2 ␣ 2 . Combine the last two equations using 2.5Ža.. ⬖ x 1 y  z 2  s x 1 yz 2  . COROLLARY. If G is a semigroup, then ␣ 12 s ␣ 1 , ␣ 22 s ␣ 2 , and G1 l G 2 s d4 where d is a right identity element for G1 and a left identity element for G 2 . Proof. The equation in Theorem 5.3 is a restricted case of the associative law. The result follows from Theorems 5.3 and 4.2. The corollary shows that the associative law puts an internal direct product into the class Žc, d, e, f. of Fig. 4.5. However it need not be in the subclass Ža, b, c, d, e, f.: Example 7.2 is a semigroup in which d is neither an identity element Žas in 4.4. nor even central Žas in 4.1.. Theorems 5.1᎐5.3 ensure that most well-known groupoids can form internal direct products only with < G1 l G 2 < s 1. Any example with < G1 l G 2 < s 0 must be infinite and without the associative law or either cancellation law. ŽPerhaps this helps to explain why Examples 7.5᎐7.9 may
613
INTERNAL DIRECT PRODUCTS
seem far-fetched.. However, the commutative law does not force < G1 l G 2 < to be 1: Example 7.6 Žwith k s 1. is commutative but has G1 l G 2 empty. Also the condition < G1 l G 2 < s 1 does not enforce any of the special properties of Theorems 5.1᎐5.3: Example 7.6 Žwith k s 0. has < G1 l G 2 < s 1 but enjoys neither finiteness nor the associative nor cancellation law.
6. CLOSER TO TRADITION Consider any internal direct product G having < G1 l G 2 < s 1, and hence belonging to the class Žc, d, e. of Fig. 4.5. Although G need not satisfy 2.6Žf. and the corresponding properties in 4.2, it is always isotopic to a groupoid which has those properties and so belongs to the class Žc, d, e, f.. If G is in Žb, c, d, e., then the isotopic groupoid is in Ža, b, c, d, e, f.. Several construction steps lead to the main result in Theorem 6.6. DEFINITIONS 6.1 Žof ␥ 11 , ␦ , 1 , 2 .. ᭙ x g G, x␥ 11 s x ␣ 12 x ␣ 22 . The binary operation ␦ s ␥ 11 . When G1 l G 2 s d4 , then ᭙ x g G, x 1 s x ␣ 1 d  , and x 2 s dx ␣ 2  . The notation ␥ 11 is part of a systematic notation ␥m n mentioned in Section 8. Theorem 6.6 shows that Ž G, ␦ , 1 , 2 . is an internal direct product of G1 and G 2 , satisfying 2.6Žc᎐f., and having Ž G, ␦ . isotopic to Ž G,  .. The proof uses several lemmas. LEMMA 6.2. ␥ 11 is an endomorphism of Ž G,  .. Proof. ᭙ x, ygG,
xy ␥ 11 s xy ␣ 12 xy ␣ 22
6.1
s x ␣ 12 y␣ 12 x ␣ 22 y␣ 22
2.5 Ž c .
s x ␣ 12 x ␣ 22 y␣ 12 y␣ 22
2.4
s x␥ 11 y␥ 11  .
6.1
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K. E. PLEDGER
LEMMA 6.3.
If G1 l G 2 s d4 , then ␥ 11 1 s 1 ␥ 11 s ␣ 1 , ␥ 11 2 s 2 ␥ 11 s ␣ 2 , G 1 s G1 ,
and
G 2 s G 2 .
Proof. ᭙ xgG,
᭙ xgG,
x␥ 11 1 s x ␣ 12 x ␣ 22␣ 1 d 
6.1
s x ␣ 12 d 
2.5 Ž b .
s x ␣1
3.4 Corollary 1
x 1 ␥ 11 s x ␣ 1 d ␣ 12 x ␣ 1 d ␣ 22
6.1
s x ␣1 d␣2 
2.5 Ž b . , ⬗ d g G 2
s x ␣ 12 d 
3.4 Ž b .
s x ␣1 .
3.4 Corollary 1
2
Similarly ␥ 11 2 s 2 ␥ 11 s ␣ 2 . ᭙ xgG,
x ␣ 1 s x␥ 11 1
from above
g G 1 . ⬖ G1 : G 1. ᭙ xgG,
x 1 s x ␣1 d g G1 .
6.1 ⬗ d g G1
⬖ G 1 : G1. ⬖ G 1 s G1. Similarly G 2 s G 2 . LEMMA 6.4. product.
If G1 l G 2 s d4 , then Ž G, ␦ , 1 , 2 . is an internal direct
Proof. The appropriate versions of 2.5Ža. ᎐ Žc. are verified as follows.
Ž a.
Ž b.
᭙ xgG,
᭙ x, ygG,
x 1 x 2 ␦ s x 1 x 2 ␥ 11
6.1
s x 1 ␥ 11 x 2 ␥ 11 
6.2
s x ␣1 x ␣2 
6.3
s x.
2.5 Ž a .
x 1 y 2 ␦ 1 s x 1 y 2 ␥ 11 1
6.1
s x 1 y 2 ␣ 1
6.3
s x1;
2.5 Ž b .
since 6.3 shows x 1 g G1 and y 2 g G 2 .
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INTERNAL DIRECT PRODUCTS
Similarly x 1 y 2 ␦ 2 s y 2 . Žc. For every x, y g G, xy ␣ 1 s x ␣ 1 y␣ 1  ⬖ xy ␥ 11 1 s x 1 ␥ 11 y 1 ␥ 11  s x 1 y 1 ␥ 11 ⬖ xy␦ 1 s x 1 y 1 ␦ .
2.5 Ž c . 6.3 6.2 6.1
Similarly xy␦ 2 s x 2 y 2 ␦ . LEMMA 6.5. If G1 l G 2 s d4 , then ␥ 11 is an automorphism of Ž G,  ., ha¨ ing in¨ erse x ¬ x 1 x 2  . Proof. ᭙ xgG,
x 1 x 2 ␥ 11 s x 1 x 2 ␦
6.1
s x,
6.4 Ž a .
and x␥ 11 1 x␥ 11 2  s x ␣ 1 x ␣ 2 
6.3
s x.
2.5 Ž a .
y1
Thus x␥ 11 s x 1 x 2  , so the endomorphism ␥ 11 of Lemma 6.2 is in fact an automorphism. Lemma 6.2 does not require that G1 l G 2 / ⭋, but Lemma 6.5 does: cf. Examples 7.5᎐7.9 and Section 8. ŽIn fact the converse of Lemma 6.5 can be proven.. THEOREM 6.6. Any internal direct product Ž G,  , ␣ 1 , ␣ 2 . with G1 l G 2 s d4 has the following properties. Ža. Ž G,  . is isotopic to Ž G, ␦ .. Žb. Ž G, ␦ , 1 , 2 . is an internal direct product of G1 and G 2 . Žc. 12 s 1 and 22 s 2 . Žd. d is a right identity element for Ž G1 , ␦ . and a left identity element for Ž G 2 , ␦ .. Proof.
Ž a.
␥ 11 is a permutation of G
6.5
and ᭙ x, y g G,
xy ␥ 11 s xy␦ .
6.1
⬖ Ž1 G , 1 G , ␥ 11 . is an isotopism from Ž G,  . onto Ž G, ␦ .. ŽLemma 6.5 shows that Ž␥ 11 , ␥ 11 , 1 G . is another..
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K. E. PLEDGER
Žb. Lemma 6.3 shows G 1 s G1 follows from Lemma 6.4.
Ž c . ᭙ xgG,
G 2 s G 2 . The result then
and
x 12 s x ␣ 1 d ␣ 1 d 
6.1
s x ␣1 d
2.5 Ž b . , ⬗ d g G2
s x 1.
6.1
Similarly 22 s 2 . Žd. now follows from Theorem 4.2.
7. EXAMPLES In each of the finite examples 7.1᎐7.4, G 2 is the opposite of G1. The infinite examples begin with 7.5. EXAMPLE 7.1. An external direct product which cannot be made internal. G1
a
b
G2
f
g
a b
b a
b a
f g
g g
f f
G1 = G 2
Ž a, f .
Ž a, g .
Ž b, f .
Ž b, g .
Ž a, f . Ž a, g . Ž b, f . Ž b, g .
Ž b, g . Ž b, g . Ž a, g . Ž a, g .
Ž b, f . Ž b, f . Ž a, f . Ž a, f .
Ž b, g . Ž b, g . Ž a, g . Ž a, g .
Ž b, f . Ž b, f . Ž a, f . Ž a, f .
G1 = G 2 has no non-trivial subgroupoids at all, so it is certainly not an internal direct product of two subgroupoids. Section 1 referred to this difficulty. As mentioned at the end of Section 3,
a s b s the transposition Ž a b . s an automorphism of G1 and
f s g s the transposition Ž f g . s an automorphism ofG 2 . However none of a, b, f, g is idempotent, so Theorem 3.5 does not apply.
INTERNAL DIRECT PRODUCTS
617
EXAMPLE 7.2. A semigroup which is an internal direct product satisfying 2.6 Žc, d, e, f. but neither Ža. nor Žb.. G1
d
a
G2
d
f
d a
d a
d a
d f
d d
f f
Here dd  s d, d is the identity automorphism of G1 , and d is the identity automorphism of G 2 , so the 3.5 conditions are satisfied. Hence combine G1 and G 2 as described after Theorem 3.5, writing t for af  . That gives: x x ␣1 x ␣2
d d d
a a d
f d f
t a f
Thus, for example, tf ␣ 1 s t ␣ 1 f ␣ 1 
2.5 Ž c .
s ad  s a
from tables above
and tf ␣ 2 s t ␣ 2 f ␣ 2  s ff  s f so tf  s tf ␣ 1 tf ␣ 2 
2.5 Ž a .
s af  s t. Calculating by components in that way leads to this table: G
d
a
f
t
d a f t
d a d a
d a d a
f t f t
f t f t
Of course the last two tables could have been given in the first place, but then they would need to be checked for conditions 2.5Ža. ᎐ Žc.. The G1 and G 2 tables occur within the G table. The table for ␣ 1 and ␣ 2 attests 2.6Žf.: ␣ 1 < G 1 and ␣ 2 < G 2 are identity mappings. Theorem 4.2 restates this in terms of d. Indeed, G1 has d Žand
618
K. E. PLEDGER
also a. as a right but not left identity element, and G 2 has d Žand also f . as a left but not right identity element. But neither d nor anything else is even a one-sided identity element for G itself, so 2.6Ža. is false. Also a g G1 and d g G 2 but a does not commute with d, so 2.6Žb. is false. Since G1 and G 2 are obviously associative, G is also. Finally, G1 alone Žor G 2 alone. illustrates the introductory paragraph of the proof of Theorem 3.5. Since dd  s d and d is an automorphism of G1 , the groupoid G1 can be seen as the internal direct product of itself and d4 . EXAMPLE 7.3. A commutative quasigroup which is an internal direct product satisfying 2.6Žb, c, d, e. but neither Ža. nor Žf..
Ž G1 ,  .
d
a
b
Ž G2 ,  .
d
f
g
d a b
d b a
b a d
a d b
d f g
d g f
g f d
f d g
Here d < G 1 is the transposition Ž a b . which is easily seen to be an automorphism of G1. Likewise d < G 2 is an automorphism of G 2 , and also dd  s d. Hence Theorem 3.5 applies, so G1 and G 2 can be combined as in the previous example to give a groupoid G of order 9, not tabulated here. By Theorem 3.4 Corollary 1, ␣ 1 < G 1 is the inverse of d < G 1 so it also is Ž a b ., and similarly ␣ 2 < G 2 s Ž f g .. Now, d commutes with every element, so Theorem 4.1 ensures 2.6Žb.: every element of G1 commutes with every element of G 2 . As an example of the Theorem 4.1 proof, ga  s df  bd  s db fd  s bd  df  s ag  . But neither d nor anything else is even a one-sided identity element, so 2.6Ža. is false. Also ␣ 1 < G 1 and ␣ 2 < G 2 as given above are not identity mappings, so 2.6Žf. is false. Incidentally, G is a finite commutative quasigroup with G1 ( G 2 , but nevertheless it satisfies neither 2.6Ža. nor Žf.. This is a convenient example to illustrate Section 6. Using Definition 6.1, ␥ 11 turns out to be the permutation Ž a b .Ž f g .Ž af  bg  .Ž ag  bf  . leaving d invariant. Hence ␦ s ␥ 11 gives, in particular, new tables for G1 and G 2 :
Ž G1 , ␦ .
d
a
b
Ž G2 , ␦ .
d
f
g
d a b
d a b
a b d
b d a
d f g
d f g
f g d
g d f
619
INTERNAL DIRECT PRODUCTS
These are both groups, so their direct product Ž G, ␦ . is just the elementary Abelian group of order 9. Other details from Theorem 6.6 are easy to check. EXAMPLE 7.4. A finite quasigroup which is an internal direct product satisfying 2.6Žc, d, e. but neither Ža., Žb., nor Žf.. G1
d
a
b
c
G2
d
f
g
h
d a b c
d b c a
c a d b
a c b d
b d a c
d f g h
d h f g
g f h d
h d g f
f g d h
Here d < G 1 is the cycle Ž a b c . which is an automorphism of G1. Likewise d < G 2 is an automorphism of G 2 , and also dd  s d. Hence Theorem 3.5 applies, so G1 and G 2 can be combined as in Example 7.2 to give a groupoid G of order 16, not tabulated here. By Theorem 3.4 Corollary 1, ␣ 1 < G 1 is the inverse of d < G 1 so it is Ž a c b ., and similarly ␣ 2 < G 2 s Ž f h g .. G is a finite cancellation groupoid Žbecause both G1 and G 2 are., and indeed satisfies 2.6Žc᎐e. as Theorems 5.1 and 5.2 require. But it has not even a one-sided identity element, so 2.6Ža. is false. Also d commutes with no other element, so 2.6Žb. is false. Also ␣ 1 < G 1 and ␣ 2 < G 2 as given above are not identity mappings, so 2.6Žf. is false. Like the previous example, this one can be used to illustrate Theorem 6.6, although in this case Ž G, ␦ . is not a group. EXAMPLES 7.5. The simplest internal direct products having G1 l G 2 s ⭋, i.e., not satisfying 2.6Žc, d, e.. Theorem 5.1 shows that any such G must be infinite, so at least one of G1 and G 2 is infinite. Hence the smallest possible orders < G1 < and < G 2 < are / 0 and 1 Žor vice versa., as in these examples. G s ⺞ j 0 4 s 0, 1, 2, . . . 4 . ᭙ m, n g G,
¡ ¢
my1 mn  s m m q Ž n y m.
~
if m ) n if m s n, if m - n
where is any mapping ⺞ ª ⺞ j 04 . ŽDifferent choices of give different examples.. ᭙ n g G,
n ␣ 1 s n q 1 and
n ␣ 2 s 0.
620
K. E. PLEDGER
Since cases with G1 l G 2 s ⭋ are unfamiliar, 2.5Ža. ᎐ Žc. are here verified in full.
Ž a. ᭙ ngG,
n ␣ 1 n ␣ 2  s Ž n q 1. 0  s Ž n q 1. y 1
⬗ nq1)0
s n. Žb. ᭙ m, n g G,
m ␣ 1 n ␣ 2 ␣ 1 s Ž m q 1 . 0 ␣ 1 s m ␣ 1
and m ␣ 1 n ␣ 2 ␣ 2 s 0 s n ␣ 2 .
Ž c.
m)n
«
m ␣ 1 n ␣ 1  s Ž m q 1. Ž n q 1.  s m s Ž m y 1 . ␣ 1 s mn ␣ 1
msn
«
m ␣ 1 n ␣ 1  s m ␣ 1 s mn ␣ 1
m-n
«
m ␣ 1 n ␣ 1  s Ž m q 1. Ž n q 1.  s m q 1 q Ž n y m . s mn  q 1 s mn ␣ 1
Also ᭙ m, n g G, m ␣ 2 n ␣ 2  s 00  s 0 s mn ␣ 2 . Hence G is indeed an internal direct product with G1 s G␣ 1 s ⺞ and G 2 s G␣ 2 s 04 , so < G1 < s / 0 and < G 2 < s 1. Incidentally also G1 j G 2 s G, showing another sense in which these examples are small. Clearly G1 l G 2 s ⭋, so 2.6Že. is false. In fact G is easily seen to satisfy 2.6Žc. but neither Ža., Žd., Že., nor Žf.. It satisfies 2.6Žb. if and only if is the mapping n ¬ n y 1 Žas in Example 7.6 with k s 1.. It follows from Theorems 5.1᎐5.3 that G is neither finite nor a cancellation groupoid nor a semigroup. Indeed, 00  s 0 s 10  and 20  s 1 s 21  are counter-examples to the two cancellation laws, and 21  0  s 10  s 0 / 1 s 20  s 210  is a counter-example to the associative law. Since G1 ␣ 1 s 2, 3, 4, . . . 4 / G1 , the conditions 3.3Ža. ᎐ Žd. are not satisfied. However, they are satisfied when the subscripts 1 and 2 are transposed, since G 2 ␣ 2 s 04 s G 2 . Section 6 mentioned that although Lemma 6.2 showed ␥ 11 to be an endomorphism in all cases, Lemma 6.5 only showed it to be an automorphism in cases where G1 l G 2 / ⭋. In the present examples, n␥ 11 s n ␣ 12 n ␣ 22 s Ž n q 2.0  s n q 1, so indeed ␥ 11 is not surjective. ŽSimilar remarks apply in Examples 7.6᎐7.9.. Theorem 2.5Žc. shows that ␣ 1 is an endomorphism, and in these examples ␣ 1: n ¬ n q 1 is injective. Hence G␣ 1 , G␣ 12 , etc. are subgroupoids isomorphic to G itself, and thus internal direct products in their
621
INTERNAL DIRECT PRODUCTS
own right. In particular, G␣ 1 s 1, 2, 3, 4, . . . 4 is an internal direct product of G␣ 12 s 2, 3, 4, . . . 4 and 14 . Thus G s internal direct product of G␣ 1 and G␣ 2 s internal direct product of Ž internal direct product of G␣ 12 and 1 4 . and 0 4 . But this internal direct multiplication is not associative. There is no internal direct product of 14 and 04 ; since their external direct product has order one, whereas there are two elements to accommodate. Hence any extension of internal direct products to more than two factors requires special care. EXAMPLES 7.6. Groupoids from 7.5 which factorize as internal direct products in more than one way. With the same Ž G,  . as in Examples 7.5, can the projections ␣ 1 and ␣ 2 be different from before? This can happen if Žand in fact only if. there is some integer k G 0 such that ᭙ n G MaxŽ k, 1., n s n y k. ŽIf k G 2, the values 1 , 2 , . . . , Ž k y 1. are still arbitrary non-negative integers.. Then Ž G,  . is an internal direct product not only as in 7.5 but also in a new way where ᭙ n g G, n ␣ 1 s 0, and n ␣ 2 s n q k. The verification of 2.5Ža. ᎐ Žc. is omitted this time. G1 s G␣ 1 s 04 and G 2 s G␣ 2 s k, k q 1, k q 2, . . . 4 . Case k s 0. ᭙ n g ⺞,
n s n, so mn  s
᭙ n g G,
n ␣ 1 s 0 and
½
my1 n
if m ) n if m F n
n ␣ 2 s n.
G is the internal direct product of G1 s 04 and G 2 s G s ⺞ j 04 . Hence G1 l G 2 s 04 / ⭋, and indeed G satisfies all of 2.6Žc᎐f. as in Theorem 4.2. Observe then that a groupoid may factorize as an internal direct product in one way which satisfies 2.6Že. as well as in another way Žcf. 7.5. which does not. Case k s 1. ᭙ n g ⺞,
n s n y 1, so mn  s
᭙ n g G,
n ␣ 1 s 0 and
½
my1 m ny1
if m ) n if m s n if m - n
n ␣ 2 s n q 1.
G is the internal direct product of G1 s 04 and G 2 s ⺞.
622
K. E. PLEDGER
The formula for mn  shows that G is commutative, so the commutative law does not force G1 l G 2 to be non-empty Žas mentioned near the end of Section 5.. In particular, every element of G1 commutes with every element of G 2 , and indeed G satisfies 2.6Žb, c. but neither Ža., Žd., Že., nor Žf., so the converse of Theorem 4.1 is false. The present factorization of G as an internal direct product is just the 7.5 factorization with the subscripts 1 and 2 transposed. The commutative law makes this possible. Cases k G 1 have G1 ␣ 1 s 04 s G1 and G 2 ␣ 2 s 2 k, 2 k q 1, 2 k q 2, . . . 4 / G 2 , so they satisfy the conditions 3.3Ža. ᎐ Žd. but not the same conditions with the subscripts 1 and 2 transposed. Cases k G 2 have G / G1 j G 2 . EXAMPLES 7.7. Groupoids resembling Examples 7.5 but with G1 and G 2 both infinite. G s Ž ⺞ j 04 . = Ž ⺞ j 04 . . ᭙ m, n, p, q g ⺞ j 04 , Ž m, n.Ž p, q .  is the ordered pair whose first component
¡m y 1 s~m ¢m q Ž p y m.
and second component
¡q y 1 ¢q q Ž n y q .
s~q
2
1
if m ) p if m s p if m - p
if q ) n if q s n if q - n,
where 1 and 2 are any mappings ⺞ ª ⺞ j 04 . ŽDifferent choices of 1 and 2 give different examples.. ᭙ m, n g ⺞ j 04 ,
Ž m, n . ␣ 1 s Ž m q 1, 0 . and Ž m, n . ␣ 2 s Ž 0, n q 1 . . Verification of 2.5Ža. ᎐ Žc. is no harder than in Examples 7.5. G1 s G␣ 1 s ⺞ = 04 and G 2 s G␣ 2 s 04 = ⺞, so G1 l G 2 s ⭋. Indeed G is easily seen to satisfy 2.6Žc. but neither Ža., Žd., Že., nor Žf.. It satisfies 2.6Žb. if and only if ᭙ n g ⺞, n 1 s n 2 s n y 1, in which case G is commutative. It violates both parts of 3.4Ža., since G1 ␣ 1 / G1 and G 2 ␣ 2 / G 2 . EXAMPLES 7.8. Internal direct products in which < G1 ␣ 2 < s 1 but < G 2 ␣ 1 < is any non-zero cardinal number at all, and the other traditions also can be violated.
623
INTERNAL DIRECT PRODUCTS
Let Ž I, . be any groupoid whose subgroupoid J has identity element j. ŽThis j need not be an identity for the whole of I.. Let and be mappings ⺞ = I ª ⺞ j 04 , arbitrary except that ᭙ n g ⺞, ᭙ y g J, ny s n y 1. Then G is the set Ž⺞ j 04. = I = J. ᭙ m, n g ⺞ j 04 , ᭙ u, ¨ g I, ᭙ x, y g J,
¡Ž Ž m y n. ¨ q n, u, xy . Ž m, u, x . Ž n, ¨ , y .  s~ Ž n, u¨ , xy . ¢Ž Ž n y m. u q m, ¨ , xy .
if m ) n if m s n if m - n
Ž m, u, x . ␣ 1 s Ž m q 1, u, j . Ž m, u, x . ␣ 2 s Ž 0, x, x . . After all that, it may be heartening to observe that Examples 7.5 are the special cases having I s J s j4 and nj s n y 1, with Ž n, j, j . abbreviated to n, and nj abbreviated to n . Even so, perhaps the check of 2.5Ža. ᎐ Žc. had better not be left to the hapless reader. First,
Ž m, u, x . ␣ 1 Ž n, ¨ , y . ␣ 2  s Ž m q 1, u, j . Ž 0, y, y .  s Ž Ž m q 1 . y , u, jy . s Ž m, u, y . Thus Ž a.
Ž b.
⬗ mq1)0 ⬗ y g J.
Ž m, u, x . ␣ 1 Ž m, u, x . ␣ 2  s Ž m, u, x . Ž m, u, x . ␣ 1 Ž n, ¨ , y . ␣ 2 ␣ 1 s Ž m, u, y . ␣ 1 s Ž m q 1, u, j . s Ž m, u, x . ␣ 1 Ž m, u, x . ␣ 1 Ž n, ¨ , y . ␣ 2 ␣ 2 s Ž m, u, y . ␣ 2 s Ž 0, y, y . s Ž n, ¨ , y . ␣ 2
Ž c . From the definitions of  and ␣ 1 ,
Ž m, u, x . Ž n, ¨ , y . ␣ 1
¡Ž Ž m y n. ¨ q n q 1, u, j . ¢Ž Ž n y m. u q m q 1, ¨ , j .
s~ Ž n q 1, u¨ , j .
if m ) n if m s n if m - n
s Ž m q 1, u, j . Ž n q 1, ¨ , j . 
⬗ jj s j
s Ž m, u, x . ␣ 1 Ž n, ¨ , y . ␣ 1  .
624
K. E. PLEDGER
Ž m, u, x . Ž n, ¨ , y . ␣ 2 s Ž something, something, xy . ␣ 2 s Ž 0, xy , xy . s Ž 0, x, x . Ž 0, y, y .  s Ž m, u, x . ␣ 2 Ž n, ¨ , y . ␣ 2  . The definitions of ␣ 1 and ␣ 2 show that G1 s Ž p, u, j . : p g ⺞, u g I 4 , so G1 ␣ 2 s Ž0, j, j .4 ; and also that G 2 s Ž0, x, x . : x g J 4 , so G 2 ␣ 1 s Ž1, x, j . : x g J 4 . It follows easily that 2.6Žd. ᎐ Žf. are all false. However 2.6Ža. ᎐ Žc. depend upon the choice of I, J, , and . Conditions for 2.6Žc. are almost immediate. From the previous paragraph < G1 ␣ 2 < s 1, but the definition of  shows that Ž G 2 ␣ 1 ,  . ( Ž J, .. Thus G satisfies 2.6Žc. if and only if < J < s 1, i.e., J s j4 . However G 2 ␣ 1 can be isomorphic to any groupoid with identity, and hence of any non-zero order. This makes Examples 7.8 very different from orthodox internal direct products as in 2.6 or even in 3.4 where every member of G 2 has the same first component d. Next consider a condition which is sufficient Žand in fact necessary . for G to satisfy 2.6Žb.. Suppose ᭙ n g ⺞, ᭙ y g J, ny s n y 1. Then
Ž m q 1, u, j . Ž 0, y, y .  s Ž Ž m q 1 . y , u, jy . s Ž m, u, y . , and
Ž 0, y, y . Ž m q 1, u, j .  s Ž Ž m q 1 . y , u, yj . s Ž m, u, y . . Thus every element Ž m q 1, u, j . of G1 commutes with every element Ž0, y, y . of G 2 , so 2.6Žb. is satisfied. Now consider some conditions which are sufficient Žand in fact necessary. for G to satisfy 2.6Ža.. Suppose Ž I, . has an identity element i f J, and ᭙ n g ⺞, ni s ni s n. Then
Ž m, u, x . Ž 0, i , j .  s
½
Ž mi , u, xj . Ž 0, ui , xj .
if m ) 0 if m s 0
s Ž m, u, x . , and
Ž 0, i , j . Ž n, ¨ , y .  s
½
Ž 0, i¨ , jy . Ž ni , ¨ , jy .
s Ž n, ¨ , y . .
if 0 s n if 0 - n
INTERNAL DIRECT PRODUCTS
625
Thus G has the identity element Ž0, i, j ., so 2.6Ža. is satisfied. Notice that G1 l G 2 is empty, yet G has an identity element somewhere else. All those conditions suggest examples to separate some of the classes in Fig. 4.5. First:
Ž I, .
i
j
i j
i j
j j
with J s j 4
and ni s ni s n,
᭙ n g ⺞,
nj s n y 1.
The conditions used in defining G are fulfilled, and the foregoing discussion shows that G satisfies 2.6Ža. and Žc.. It also satisfies 2.6Žb. if ᭙ n g ⺞, nj s n y 1, which provides an example satisfying 2.6Ža᎐c. but neither Žd., Že., nor Žf.. But otherwise ᭚ n g ⺞: nj / n y 1, say, for example, 5 j s 8. In that case Ž5, i, j .Ž0, j, j .  s Ž4, i, j . / Ž8, i, j . s Ž0, j, j .Ž5, i, j .  contrary to 2.6Žb.; which provides an example satisfying 2.6Ža, c. but neither Žb., Žd., Že., nor Žf.. Next:
Ž I, .
i
j
k
i j k
i j k
j j k
k k k
with J s j, k 4 ,
and ᭙ n g ⺞,
ni s ni s n,
nj s nk s nj s nk s n y 1.
The conditions used in defining G are fulfilled, and the earlier discussion shows that G satisfies 2.6Ža, b. but neither Žc., Žd., Že., nor Žf.. Finally:
Ž I, .
j
k
j k
j k
k k
with J s I,
and ᭙ n g ⺞,
nj s nk s n y 1,
but 5 j s 8. The reader may check that G is then an internal direct product satisfying none of the six conditions 2.6Ža. ᎐ Žf.. This highly unorthodox example vindicates the claim at the end of Section 2.
626
K. E. PLEDGER
EXAMPLES 7.9. Internal direct products in which < G 2 ␣ 1 < and < G1 ␣ 2 < are any non-zero cardinal numbers. The construction details will not be given. But just as each Example 7.7 is compounded from two versions of 7.5, so each Example 7.9 can be constructed analogously from two versions of 7.8. The elements are now ordered sextuples so the notation looks intricate, but the generalization from Examples 7.8 raises no other difficulties.
8. EPILOGUE Examples 7.5᎐7.9 perhaps look like conjuring tricks. It seems only fair to disclose briefly how the rabbits got into the hats. Consider any internal direct product Ž G,  , ␣ 1 , ␣ 2 .. For all non-negative integers m and n, define ␥m n : G ª G by x␥m n s x ␣ 1mq 1 x ␣ 2nq 1 . Every such ␥m n is an endomorphism of G Žas shown in Theorem 2.5Ža. for ␥ 00 and in Lemma 6.2 for ␥ 11 .. If Gm n is defined as G␥m n , then Gm n l Gp q s G Ma xŽ m, p., MaxŽ n, q. . This can be extended by defining G⬁0 s F Gm 0 : m g ⺞4 and G 0⬁ s F G 0 n : n g ⺞4 and then ᭙ r, s g 04 j ⺞ j ⬁4 , Gr s s Gr 0 l G 0 s . Incidentally the Theorem 3.3 conditions are also equivalent to G10 s G, the same conditions with transposed subscripts are equivalent to G 01 s G, and the Theorem 3.4 conditions are equivalent to G10 s G s G 01 . G can now be partitioned into four subsets as follows. Define K s G y Ž G10 j G 01 ., L1 s G⬁0 y G 01 , and L2 s G 0⬁ y G10 . Also write ⌫ s ␥m n : m, n g ⺞ j 044 , so, for example, K ⌫ s x␥m n : x g K, m G 0, n G 04 . Then G is the disjoint union K ⌫ j ˙ L1 ⌫ j ˙ L2 ⌫ j ˙ G⬁⬁. The Theorem 3.4 conditions such as < G1 l G 2 < s 1 are equivalent to K s L1 s L2 s ⭋, i.e., G s G⬁⬁ . ŽLemma 6.5 partly proves this.. Examples 7.5 and 7.8 have K s L1 s G⬁⬁ s ⭋, so G s L2 ⌫. Examples 7.5 were constructed by writing L2 s 04 and each 0␥m n s m Žand in fact include isomorphic copies of all groupoids L2 ⌫ having < L2 < s 1.. After a long struggle, Examples 7.8 were constructed by writing L2 s 04 = I = J and each Ž0, u, x .␥m n s Ž m, u, x .. Examples 7.7 and 7.9 have L1 s L2 s G⬁⬁ s ⭋, so G s K ⌫. Examples 7.7 were constructed by writing K s Ž0, 0.4 and each Ž0, 0.␥m n s Ž m, n.. Examples 7.9 have K s 04 = I1 = J1 = 04 = I2 = J 2 . An obvious final question Žvital for the refinement property of w5x etc.. is how to generalize this whole theory of internal direct products to more than the two factors G1 and G 2 . The ideal would be to find minimum restrictions permitting such a generalization. Some restrictions would probably be needed because Examples 7.5 display a non-associative aspect of internal direct multiplication, and perhaps also because the related
INTERNAL DIRECT PRODUCTS
627
Example 7.6 Žwith k s 0. has one factorization which satisfies the conditions 2.6Žc᎐e. and another which does not.
REFERENCES 1. P. M. Cohn, ‘‘Universal Algebra,’’ Harper & Row, 1965; Reidel, 1981. 2. P. Crawley and B. Jonsson, Refinements for infinite direct decompositions, Pacific J. Math. ´ 14 Ž1964., 797᎐855. 3. J. M. G. Fell and A. Tarski, On algebras whose factor algebras are Boolean, Pacific J. Math. 2 Ž1952., 297᎐318. 4. B. Jonsson, The unique factorization problem for finite relational structures, Colloq. Math. ´ 14 Ž1966., 1᎐32. 5. B. Jonsson and A. Tarski, ‘‘Direct Decompositions of Finite Algebraic Systems,’’ Notre ´ Dame Mathematical Lectures No. 5, Notre Dame, Indiana, 1947. 6. J. Łos, ´ Direct sums in general algebra, Colloq. Math. 14 Ž1966., 33᎐38. 7. R. Willard, Varieties having Boolean factor congruences, J. Algebra 132 Ž1990., 130᎐153.