- Email: [email protected]
On ∗–λ-semirings
Information Sciences 177 (2007) 5012–5023 www.elsevier.com/locate/ins
On *–k-semirings Feng Feng a
a,*
, Young Bae Jun b, Xian Zhong Zhao
c
Department of Applied Mathematics and Applied Physics, Xi’an Institute of Posts and Telecommunications, Xi’an 710061, PR China b Department of Mathematics Education, Gyeongsang National University, Chinju 660-701, South Korea c Department of Mathematics, Northwest University, Xi’an 710069, PR China Received 10 October 2006; received in revised form 11 April 2007; accepted 6 May 2007
Abstract A *–k-semiring is an ordered semiring S equipped with a star operation such that for any a, b 2 S, a*b is the least fixed point of the linear mapping x # ax + b over S. The notion of *–k-semirings is a generalization of several important concepts such as continuous semirings, (weak) inductive *-semirings and the Kleene algebras of Conway and Kozen. We investigate several basic properties of *–k-semirings and obtain results on *–k-semirings in relation to preorders, duality and formal power series. Some of these results can be seen as generalizations of relevant results on inductive *-semirings. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Semiring; *–k-semiring; Preorder; Formal power series
1. Introduction It is well known that fixed points play a fundamental role in computer science. For example, the least and the greatest fixed points of an order preserving function have shown to be the basis of logic of programs. Moreover, context free grammars are indeed systems of fixed point equations over the semiring of languages. All of these facts justify the various attempts devoted to the study of algebraic structures equipped with fixed point operations. On the other hand, semirings have been found useful for solving problems in different areas of applied mathematics and information sciences, since the structure of a semiring provides an algebraic framework for modelling and studying the key factors in these applied areas. The applications of semirings to areas such as optimization theory, graph theory, theory of discrete event dynamical systems, generalized fuzzy computation, automata theory, formal language theory, coding theory and analysis of computer programs have been extensively studied in the literature (cf. [8,9,11]). ´ sik and Kuich [4] defined an inductive *-semiring to be a partially ordered semiring equipped Recently, E with a star operation satisfying the fixed point equation and the fixed point induction rule for linear terms. *
Corresponding author. E-mail addresses: [email protected] (F. Feng), [email protected] (Y.B. Jun), [email protected] (X.Z. Zhao).
0020-0255/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2007.05.019
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They showed that any inductive *-semiring is a Conway semiring and an iteration semiring satisfying 1* = 1**. In addition, they proved that both the power series semiring and the matrix semiring of any inductive *-semiring are inductive *-semirings. Following this line of exploration, Feng et al. [7] introduced *–k-semirings and proved that if S is a weak inductive *-semiring, so is the semiring of power series ShhA*ii. This gave a positive answer to one of E´sik and Kuich’s open problems in [4]. The notion of *–k-semirings extends several important concepts such as continuous semirings, (weak) inductive *-semirings and the Kleene algebras [3,10,12] of Conway and Kozen. As a continuation of our previous work [7], we study several basic properties of *–k-semirings in the present paper and obtain results on *–k-semirings relevant to preorders, duality and formal power series. Some of these results can be seen as generalizations of relevant results on (weak) inductive *-semirings. The paper is organized as follows. Section 2 gives some basic concepts and results used for our discussion. In Section 3, we first recall our definitions of *–l-semirings and *–k-semirings which were proposed in [7]. Then we give an equivalent characterization of *–k-semirings and present the basic equations or inequations which hold in any *–k-semiring. In Section 4, we define natural F-preorders on a monoid and study F-sum-preorders on semirings. We introduce the notion of F-sum-ordered semirings which is a straightforward generalization of sum-ordered semirings discussed in [4,13]. We show that the notions of (weak) inductive *-semirings, *–l-semirings and *–k-semirings are equivalent concepts for F-sum-ordered semirings. Section 5 is devoted to duality. We introduce and investigate symmetric *–k-semirings, symmetric *–l-semirings and symmetric weak inductive *-semirings. We discuss some of the relationships among these *-semirings. In addition, we consider the semiring of power series and prove that if S is a symmetric *–k-semiring [symmetric weak inductive *-semiring], so is the semiring of power series ShhA*ii. 2. Basic notions A semiring is an algebra S = (S, +, Æ, 0, 1) equipped with binary operations + (sum or addition) and Æ (product or multiplication) and constants (or nullary operations) 0 and 1 such that the additive reduct (S, +, 0) is a commutative monoid with identity 0, the multiplicative reduct (S, Æ, 1) is a monoid with identity 1 and multiplication distributes over all finite sums, including the empty sum. That is ða þ bÞc ¼ ac þ bc; cða þ bÞ ¼ ca þ cb; a0 ¼ 0a ¼ 0 hold for all a, b, c 2 S. A morphism of semirings is a function that preserves the operations and constants. A semiring S is said to be ordered if it is equipped with a partial order such that the operations on S are monotonic. A morphism of ordered semirings also preserves the partial order. A *-semiring is a semiring S equipped with a star operation * : S ! S. Morphisms of *-semirings also preserve the star operation. An ordered *-semiring means an ordered semiring equipped with a star operation (where the star operation need not be monotonic). A morphism of ordered *-semirings is an order preserving *-semiring morphism. An ordered *-semiring is called an inductive *-semiring [4] if it satisfies the fixed point inequation aa þ 1 6 a
ð1Þ
and the fixed point induction rule ax þ b 6 x ) a b 6 x:
ð2Þ
E´sik and Kuich [4] proved that the following equations: aa þ 1 ¼ a ; a a þ 1 ¼ a ; ðabÞ ¼ 1 þ aðbaÞ b; ðabÞ a ¼ aðbaÞ ; ða þ bÞ ¼ ða bÞ a hold in any inductive *-semiring and the star operation on any inductive *-semiring is monotonic.
ð3Þ ð4Þ ð5Þ ð6Þ ð7Þ
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A weak inductive *-semiring [4] is an ordered *-semiring which satisfies the fixed point equation (3), the sum star equation (7) and the weak fixed point induction rule ax þ b ¼ x ) a b 6 x:
ð8Þ
Let S be a semiring. Then any mapping f of the form x # ax + b, where a, b 2 S, is called a linear mapping over the semiring S. Since both addition and multiplication on an ordered semiring are monotonic, we deduce that any linear mapping over an ordered semiring is monotonic as well. Let S be a poset and let f be a mapping on S. We say p 2 S is a prefixed point of f if f(p) 6 p. The least element of the set of all the prefixed points of f, when it exists, is called the least prefixed point of f and is denoted by l.f. An element p of S is called a fixed point of f if f(p) = p. The least fixed point of f is similarly defined and is denoted by k.f. If f is concretely given by terms, such as f(x) = ax + b, we write the least prefixed [fixed] point of f as lx(ax + b) [kx(ax + b), respectively]. 3. *–l-semirings and *–k-semirings In [7], the authors defined a l-semiring1 [k-semiring] to be an ordered semiring S on which every linear mapping has a least prefixed [fixed] point. They also defined a *–l-semiring [*–k-semiring] to be an ordered *-semiring S such that any linear mapping f : x # ax + b over S has a least prefixed [fixed] point lx.f = a*b [kx.f = a*b, respectively]. Tarski in [14] proved that the least prefixed point of a monotonic function f is a fixed point of f. Thus, we deduce the following. Lemma 3.1 [7]. A l-semiring S is a k-semiring such that kx.f = lx.f for any linear mapping f : x # ax + b over S. In particular, every *–l-semiring is a *–k-semiring satisfying kx:f ¼ lx:f ¼ a b for any linear mapping f : x # ax + b over S. Using Lemma 3.1 and the definitions above, we established in [7] the following result. Proposition 3.2. Suppose that S is an ordered *-semiring. Then the following conditions are equivalent: (1) (2) (3) (4)
S S S S
is is is is
an inductive *-semiring. a *–l-semiring. both a l-semiring and a *–k-semiring. both a l-semiring and a weak inductive *-semiring.
The next result gives an equivalent characterization of *–k-semirings, from which we deduce that any (weak) inductive *-semiring is a *–k-semiring. Lemma 3.3. Let S be an ordered *-semiring. Then S is a *–k-semiring if and only if S satisfies the fixed point equation (3) and the weak fixed point induction rule (8). Hence, S is a *–k-semiring satisfying (7) if and only if S is a weak inductive *-semiring. Proof. Suppose that S is a *–k-semiring. Then for any a, b 2 S, a*b is the least fixed point of the linear mapping x # ax + b over S. In particular, a* is the least fixed point of the mapping x # ax + 1 and so the fixed point equation (3) holds in S. Furthermore, let a, b, x 2 S with ax + b = x. Then x is a fixed point of the linear mapping x # ax + b. But by definition, a*b is the least fixed point of the linear mapping x # ax + b. Hence it follows that a*b 6 x and so S satisfies the weak fixed point induction rule (8).
1
´ sik and Leiß in [5]. Notice that a different notion of l-semirings is given by E
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Conversely, assume that S is an ordered *-semiring satisfying (3) and (8). Then for any linear mapping f : x # ax + b over S, we have f ða bÞ ¼ aða bÞ þ b ¼ ðaa þ 1Þb ¼ a b by the fixed point equation (3). Hence, a*b is a fixed point of f. Moreover, we easily deduce that a*b is the least fixed point of f by the weak fixed point induction rule (8). Thus S is a *–k-semiring as required. h ´ sik and Kuich [4] proved that Eqs. (4)–(6) hold in any inductive *-semiring. However, the following result E says that these equations actually hold in any *–k-semiring (therefore, they also hold in any weak inductive *-semiring). Lemma 3.4. The dual fixed point equation (4), the product star equation (5) and Eq. (6) hold in any *–k-semiring. Proof. Suppose that S is a *–k-semiring. Then by Lemma 3.3, S is an ordered *-semiring satisfying the fixed point equation (3) and the weak fixed point induction rule (8). First, we show that S satisfies the dual fixed point equation (4). In fact, since for any a 2 S aða a þ 1Þ þ 1 ¼ ðaa þ 1Þa þ 1 ¼ a a þ 1; we deduce that a* 6 a*a + 1 by (8). To prove the reverse inequation, notice that for any a, b 2 S,
abaðbaÞ þ a ¼ aðbaðbaÞ þ 1Þ ¼ aðbaÞ : Thus we have (ab)*a 6 a(ba)* by (8). Taking b = 1, we deduce that a*a 6 aa*. Then it follows that a a þ 1 6 aa þ 1 ¼ a ; proving that S satisfies (4). Next, we prove that S satisfies (5). Note first that we have proved above that S satisfies the inequation
ðabÞ a 6 aðbaÞ : Hence, it follows that
aðbaÞ b þ 1 6 abðabÞ þ 1 ¼ ðabÞ : Thus it remains to show the reverse inequation. However, this follows immediately from the fact: abðaðbaÞ b þ 1Þ þ 1 ¼ aðbaðbaÞ þ 1Þb þ 1 ¼ aðbaÞ b þ 1 by the weak fixed point induction rule (8). Finally, using Eqs. (4) and (5) proved above, we deduce
ðabÞ a ¼ ðaðbaÞ b þ 1Þa ¼ aððbaÞ ba þ 1Þ ¼ aðbaÞ and so Eq. (6) also holds in S.
h
By Lemma 3.3, we see that a *–k-semiring S is a weak inductive *-semiring if and only if S satisfies the sum star equation (7). In addition, if S is an ordered *-semiring with a commutative multiplication, we established the following result in [7]. Proposition 3.5. Let S be an ordered *-semiring with a commutative multiplication. Then S is a weak inductive *semiring if and only if S is a *–k-semiring. Proposition 3.6. Any *–k-semiring satisfies the following inequations: 0 6a; a 6a þ b; n X i¼0
ai 6a ; n P 0:
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Proof. Suppose that S is a *–k-semiring. For any a 2 S, since 1 Æ a + 0 = a, we deduce that 0 6 a by (8). Note that the second inequation is equivalent to the first one in any ordered semiring. Thus it holds in S since the first inequation holds as proved above. For the last inequation, we only need to apply the fixed point equation (3) repeatedly and as a result, we have n X a ¼ anþ1 a þ ai i¼0
for any n P 0. Then by the second inequation proved above, it follows that n n X X ai 6 anþ1 a þ ai ¼ a i¼0
i¼0
for any n P 0. h Problem 3.7. Does there exist a *–k-semiring with a noncommutative multiplication which is not a weak inductive *-semiring? Problem 3.8. Does there exist a *–k-semiring with a nonmonotonic star operation? Does there exist a *–ksemiring with a monotonic star operation which is not an inductive *-semiring? 4. F-sum-ordered semirings Recall that a preorder on a set A means a reflexive transitive binary relation on A. Let S be a monoid and F be a submonoid of S. The binary relations 6lF , 6rF and 6jF defined on S by a6lF b () ð9f 2 F Þ fa ¼ b; a6rF b () ð9f 2 F Þ af ¼ b; a6jF b () ð9f1 ; f2 2 F Þ f 1 af2 ¼ b for all a, b 2 S are preorders on S. We refer to 6lF , 6rF and 6jF as left natural F-preorder, right natural F-preorder and two-sided natural F-preorder, respectively. In general, we call them natural F-preorders on S. Natural F-preorders play an important role in diverse areas and especially in semigroup theory. To see this, we would like to point out the following examples. Example 4.1. Let S be a monoid. Then it is easy to see that the equivalence relations 6lS \ ð6lS Þ1 , 6rS \ ð6rS Þ1 and 6jS \ ð6jS Þ1 induced by the natural S-preorders are precisely the Green’s equivalence relations L, R and J on S, respectively. Example 4.2. Let S be an inverse monoid with semilattice of idempotents E. Then the natural partial order on S defined by a 6 b () ð9e 2 EÞ
a ¼ eb
is the converse of the left natural E-preorder 6lE . That is, 6¼ ð6lE Þ1 . Given a semiring S and a subsemiring F of S. Let us consider the additive reduct (S, +, 0) of S. Since F is a subsemiring of S, (F, +, 0) is a commutative submonoid of (S, +, 0). It is clear that the natural (F, +, 0)-preorders on S are all the same, i.e., 6lðF ;þ;0Þ ¼ 6rðF ;þ;0Þ ¼ 6jðF ;þ;0Þ . In this case, we simply denote the natural (F, +, 0)preorder by 6þ F and refer to it as the F-sum-preorder on S. Definition 4.3. Let S be a semiring partially ordered by the relation 6 and F be a subsemiring of S. Then S is called a F-sum-ordered semiring if 6þ F ¼6, i.e., when a 6 b () ð9f 2 F Þ for all a, b 2 S.
b¼f þa
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In particular, if the subsemiring F is the semiring S itself, then the S-sum-ordered semiring coincides with the sum-ordered semiring discussed in [4,13]. Proposition 4.4. Suppose that S is a semiring partially ordered by the relation 6 and F is a subsemiring of S. Then the following conditions are equivalent: (1) 0 is the least element of F, i.e., 0 6 f holds for all f 2 F. (2) For all f, g 2 F, g 6 g + f. (3) 6þ F is a partial order on S included in 6. Proof. It is clear that the first two conditions are equivalent in any ordered semiring. So we only need to show that the first condition implies the third and vice versa. Assume that 0 is the least element of F. If a6þ F b then f + a = b for some f 2 F. Thus it follows that a = 0 + a 6 f + a = b. That is, 6þ is included in 6. Moreover, we easily deduce that F þ 6þ F \ ð6F Þ
1
ð6 \61 Þ ¼ DS ;
þ where DS refers to the identity relation on S. But 6þ F is a preorder, it follows that 6F is a partial order on S included in 6. þ Now suppose that 6þ F is a partial order on S included in 6. For all f 2 F, we have 06F f since 0 + f = f. But þ by assumption, 6F 6. Thus it follows that 0 is the least element of F. h
Proposition 4.5. Let S and S 0 be ordered semirings. Let F be a subsemiring of S and h : S ! S 0 be a semiring morphism. If S is F-sum-ordered and 0 is the least element of h(F), then h is an ordered semiring morphism. Proof. Suppose that a 6 b in S. Then there exists f 2 F such that f + a = b since S is F-sum-ordered. By assumption, h is a semiring morphism and so h(F) is a subsemiring of S 0 , in which we have h(f) + h(a) = h(b). 0 Hence we deduce hðaÞ6þ hðF Þ hðbÞ in S . Moreover, by hypothesis, 0 is the least element of h(F). Thus by Proposition 4.4, it follows that h(a) 6 h(b) and so we conclude that h is an ordered semiring morphism. h The following two results are immediate consequences of the above propositions. Corollary 4.6. Suppose that S is a semiring partially ordered by the relation 6. Then S is a positive semiring, i.e., þ 0 is the least element of S if and only if 6þ S 6 and S is sum-ordered with respect to 6S . Corollary 4.7 [4]. Suppose that S and S 0 are ordered semirings and h : S ! S 0 is a semiring morphism. If S is sum-ordered and 0 is the least element of S 0 , then h is an ordered semiring morphism. To our knowledge, whether the fixed point induction rule (2) and the weak fixed point induction rule (8) are equivalent conditions is an interesting problem which is still open. However, as indicated by the following result, these two conditions are actually equivalent for any F-sum-ordered semiring. Lemma 4.8. Let S be an ordered *-semiring and F be a subsemiring of S. If S is F-sum-ordered, then S satisfies the fixed point induction rule (2) if and only if S satisfies the weak fixed point induction rule (8). Proof. It is clear that if S satisfies the fixed point induction rule (2), then S satisfies the weak fixed point induction rule (8). Thus we only need to show that if S is a F-sum-ordered semiring satisfying (8), then S also satisfies (2). Suppose that a, b, x 2 S with ax + b 6 x. Since S is a F-sum-ordered semiring, there exists f 2 F such that (ax + b) + f = x. Hence, we have a ðb þ f Þ 6 x by (8). But b 6 b + f since S is F-sum-ordered. By monotonicity, it follows that a b 6 a ðb þ f Þ 6 x and we conclude that S satisfies the fixed point induction rule (2). h
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The following result says that the notions of (weak) inductive *-semirings, *–l-semirings and *–k-semirings are equivalent concepts for F-sum-ordered semirings. Theorem 4.9. Suppose that S is an ordered *-semiring and F is a subsemiring of S. If S is F-sum-ordered, then the following conditions are equivalent: (1) (2) (3) (4)
S S S S
is is is is
an inductive *-semiring. a *–l-semiring. a weak inductive *-semiring. a *–k-semiring.
Proof. First, for any ordered *-semiring S (needless to be F-sum-ordered), we have that (1) ) (2) ) (3) ) (4) by Proposition 3.2 and Lemma 3.3. Thus it remains to show that if S is a *–k-semiring which is F-sum-ordered, then S is an inductive *-semiring. So suppose that S is a F-sum-ordered *–k-semiring. Then by Lemma 3.3, we know that S satisfies the fixed point equation (3) and the weak fixed point induction rule (8). Thus it follows that S satisfies the fixed point in Eq. (1). Moreover, by Lemma 4.8, we deduce that S satisfies the fixed point induction rule (2) since it satisfies the weak fixed point induction rule (8). Hence we conclude that S is an inductive *-semiring. h As an immediate consequence derived from the above result, we have Corollary 4.10 [4]. Suppose that S is a *-semiring which is a sum-ordered semiring. Then S is an inductive *-semiring if and only if S satisfies the fixed point (in)equation and the weak fixed point induction rule. Thus, S is an inductive *-semiring if and only if S is a weak inductive *-semiring. 5. Duality E´sik and Kuich [4] introduced the following concepts of duality. Suppose that S is a *-semiring. The dual (or opposite) semiring of S is a *-semiring Sop equipped with the same operations and constants as S except for multiplication, which is the reverse of the multiplication on S. When S is ordered, so is Sop equipped with the same partial order. It is clear that (Sop)op = S. The dual top of a *-semiring term t is defined by induction on the structure of t as follows: w w w w
If If If If
t is a variable or one of the constants 0, 1, then top = t. op t = t1 + t2 then top ¼ top 1 þ t2 . op op op t = t1t2 then t ¼ t2 t1 . t ¼ t1 then top ¼ ðtop 1 Þ .
op Thus by definition, we have (top)op = t. The dual of an equation t1 = t2 is the equation top 1 ¼ t 2 and the dual op op op op of an implication (t1 = s1) ^ ^ (tn = sn) ) t = s is the implication ðt1 ¼ s1 Þ ^ ^ ðtn ¼ sn Þ ) top ¼ sop . The dual of an inequation t 6 s or implication (t1 6 s1) ^ ^ (tn 6 sn) ) t 6 s is defined in the same way. It is easy to see that the dual fixed point equation (4) is indeed the dual of the fixed point equation (3). Similarly, the dual fixed point induction rule
xa þ b 6 x ) ba 6 x
ð9Þ
is the dual of the fixed point induction rule (2). Moreover, an (in)equation (or implication) is said to be self dual if its dual is equivalent to itself. The product star equation is an example of a self dual equation. By definition of duality, it is easy to prove the following. Lemma 5.1 [4]. The dual of an (in) equation or implication holds in a *-semiring S if and only if it holds in Sop. A Conway semiring [1,2,4,6] is a *-semiring S which satisfies the product star equation (5) and the sum star equation (7). By Proposition 36 of [4], the dual of a Conway semiring is a Conway semiring. However, by the following example given by Kozen [10], we claim that the corresponding fact does not hold for *–k-semirings.
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Example 5.2. Let X be an infinite set. Let F be the set of all strict, finitely additive set functions f : PðX Þ ! PðX Þ, i.e., functions f satisfying f ðA [ BÞ ¼ f ðAÞ [ f ðBÞ and f(;) = ;. When f ; g 2 F, define for any A 2 PðX Þ ðf þ gÞðAÞ ¼ f ðAÞ [ gðAÞ; ðf gÞðAÞ ¼ f ðgðAÞÞ: Moreover, let the constants in F be functions 0; 1 2 F such that 0ðAÞ ¼ ;; 1ðAÞ ¼ A for all A 2 PðX Þ. And the star operation on F is given by f ðAÞ ¼ [a f a ðAÞ for all A 2 PðX Þ, where f 0 ðAÞ ¼ A; f aþ1 ðAÞ ¼ f ðf a ðAÞÞ and f k ðAÞ ¼ [a
ð10Þ
Proposition 5.7. A weak inductive *-semiring S is a symmetric weak inductive *-semiring if and only if Sop is a weak inductive *-semiring.
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Proof. Suppose that S is a weak inductive *-semiring. If S is a symmetric weak inductive *-semiring, then S satisfies the dual weak fixed point induction rule (10). Thus by Lemma 5.1, Sop satisfies the weak fixed point induction rule (8). But by Corollary 10 of [4], any weak inductive *-semiring is a Conway semiring. Hence, S is a Conway semiring and it follows that Sop is also a Conway semiring by Proposition 36 of [4]. Thus Sop is a Conway semiring satisfying (8). Moreover, it is easy to verify that in any *-semiring, the product star equation (5) implies the fixed point equation (3). Thus we conclude that Sop is an ordered *-semiring satisfying (3), (7) and (8). Hence, Sop is a weak inductive *-semiring as required. Conversely, if Sop is a weak inductive *-semiring, then S satisfies the dual weak fixed point induction rule (10) by Lemma 5.1. By hypothesis, S is a weak inductive *-semiring. Hence, it follows that S is a symmetric weak inductive *-semiring. h By Lemma 3.3, an ordered *-semiring is a *–k-semiring if and only if it satisfies the fixed point equation (3) and the weak fixed point induction rule (8). Considering duality, we obtain the following result for symmetric *–k-semirings. Lemma 5.8. An ordered *-semiring S is a symmetric *–k-semiring if and only if S satisfies the fixed point equation (3), the weak fixed point induction rule (8) and their duals (4) and (10). Now we establish the next proposition from the above results. Proposition 5.9. Suppose that S is an ordered *-semiring. Then the following conditions are equivalent: (1) S is a symmetric *–k-semiring. (2) S is a *–k-semiring such that Sop is a *–k-semiring. (3) S is a *–k-semiring satisfying the dual weak fixed point induction rule.
Proof. By Lemmas 3.3, 5.1 and 5.8, it is easy to see that the first condition implies the second and the second condition implies the third. It remains to show that the last condition implies the first. So suppose that S is a *– k-semiring satisfying the dual weak fixed point induction rule (10). By Lemma 3.4, we know that S satisfies the dual fixed point equation (4). But by Lemma 3.3, S also satisfies the fixed point equation (3) and the weak fixed point induction rule (8). Thus, it follows that S is a symmetric *–k-semiring by Lemma 5.8. h By Lemma 3.3 and Proposition 5.9, one easily proves that any symmetric weak inductive *-semiring is a symmetric *–k-semiring. Moreover, if the multiplication on S is commutative, we have the following symmetric version of Proposition 3.5. Proposition 5.10. Let S be an ordered *-semiring with a commutative multiplication. Then S is a symmetric *–k-semiring if and only if S is a symmetric weak inductive *-semiring. Proof. As mentioned above, it is easy to prove that if S is a symmetric weak inductive *-semiring, then S is a symmetric *–k-semiring by Lemma 3.3 and Proposition 5.9. It remains to show that if S is a symmetric *–k-semiring with a commutative multiplication, then S is a symmetric weak inductive *-semiring. So suppose that S is a commutative symmetric *–k-semiring. Since S is a symmetric *–k-semiring, we immediately have that S is a *–k-semiring and so S is a weak inductive *-semiring by Proposition 3.5. But by Proposition 5.9, S also satisfies the dual weak fixed point induction rule (10). Hence, we conclude that S is a symmetric weak inductive *-semiring. h By the preceding results, Proposition 3.2 transfers to the symmetric case. Theorem 5.11. Let S be an ordered *-semiring. Then the following conditions are equivalent: (1) (2) (3) (4)
S S S S
is is is is
a a a a
symmetric symmetric symmetric symmetric
inductive *-semiring. *–l-semiring. l-semiring and a symmetric *–k-semiring. l-semiring and a symmetric weak inductive *-semiring.
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Proof. (1. ) 2.) Suppose that S is a symmetric inductive *-semiring. Then S satisfies the fixed point equation (3), the fixed point induction rule (2) and their duals (4) and (9). Thus for any linear mapping f : x # ax + b over S, we have lx:f ¼ a b
ð11Þ
by (2) and (3). Dually, by (4) and (9), we have lx:f op ¼ ba :
ð12Þ
Clearly, it follows from (11) and (12) that S is a symmetric *–l-semiring. (2. ) 3.) Suppose that S is a symmetric *–l-semiring. Then by definition, S is a symmetric l-semiring. For any linear mapping f : x # ax + b over S, by Lemma 3.1, we have kx:f ¼ lx:f ¼ a b and by duality, we have kx:f op ¼ lx:f op ¼ ba : Hence, S is both a symmetric l-semiring and a symmetric *–k-semiring. (3. ) 4.) Assume that S is both a symmetric l-semiring and a symmetric *–k-semiring. We need to show that S is a symmetric weak inductive *-semiring. In fact, since S is a symmetric l-semiring, we know that both lx(ax + b) and lx(xa + b) exist for any a, b 2 S. By assumption, S is also a symmetric *–k-semiring and so we have kx(ax + b) = a*b and kx(xa + b) = ba* for any a, b 2 S. Hence by Lemma 3.1, we deduce that lxðax þ bÞ ¼ kxðax þ bÞ ¼ a b
ð13Þ
and dually we have lxðxa þ bÞ ¼ kxðxa þ bÞ ¼ ba
ð14Þ
for any a, b 2 S. From (13), a* is the least prefixed point of the linear mapping x # ax + 1 and so we deduce that the fixed point inequation (1) holds in S. Similarly, we easily deduce that the fixed point induction rule (2) holds in S by (13). Thus S is an inductive *-semiring and so a weak inductive *-semiring. Furthermore, it follows from (14) that S satisfies the dual (weak) fixed point induction rule. Hence, we conclude that S is both a symmetric lsemiring and a symmetric weak inductive *-semiring. (4. ) 1.) Assume that S is both a symmetric l-semiring and a symmetric weak inductive *-semiring. Then it is clear that S is a weak inductive *-semiring and so S is a *–k-semiring by Lemma 3.3. Since S also satisfies the dual weak fixed point induction rule (10), we deduce that S is a symmetric *–k-semiring by Proposition 5.9. Hence S is both a symmetric l-semiring and a symmetric *–k-semiring. Consequently, as shown in (3. ) 4.), S is an inductive *-semiring satisfying the dual fixed point induction rule (9). Hence, we conclude that S is a symmetric inductive *-semiring. h In the end of this section, we discuss the semirings of formal power series. Let S be a semiring and A be a set. A formal power series over A with coefficients in S is a function r : A ! S; usually denoted by X ðr; uÞu; r¼ u2A
where (r, u) is the value r(u) of function r on the word u, and A* is the free monoid of all words over A including the empty word . Equipped with the operations of pointwise sum and Cauchy product, the set of all formal power series over A with coefficients in S forms a semiring, which is called the formal power series semiring of S and is denote by ShhA*ii. If S is ordered, ShhA*ii is converted into an ordered semiring by the pointwise order. Suppose that S is a *-semiring. For any r 2 ShhA*ii and any u 2 A*, if u = , we define ðr ; Þ ¼ ðr; Þ ;
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otherwise, define
ðr ; uÞ ¼ ðr; Þ
X
ðr; vÞðr ; wÞ:
vw¼u;v6¼
Then we have a star operation on ShhA*ii. ´ sik and Kuich proved that if S is an inductive *-semiring, then so is the formal power series semiring In [4], E * ShhA ii. For *–k-semirings and weak inductive *-semirings, we established in [7] the following results. Theorem 5.12. If S is a *–k-semiring, then so is ShhA*ii. Theorem 5.13. If S is a weak inductive *-semiring, then so is ShhA*ii. To show that the formal power series semiring of any symmetric *–k-semiring is again a symmetric *–ksemiring, we need the following auxiliary result. Proposition 5.14 [4]. Suppose that S is a *-semiring and A is a set. Then the semirings (ShhA*ii)op and SophhA*ii are isomorphic. Theorem 5.15. If S is a symmetric *–k-semiring, then so is ShhA*ii. Proof. Assume that S is a symmetric *–k-semiring. Then by Proposition 5.9, S is a *–k-semiring such that Sop is also a *–k-semiring. Thus by Theorem 5.12, we have that the formal power series semirings ShhA*ii and SophhA*ii are *–k-semirings. But by Proposition 5.14, SophhA*ii is isomorphic to (ShhA*ii)op. Hence, it follows that ShhA*ii is a *–k-semiring such that (ShhA*ii)op is also a *–k-semiring. Thus by Proposition 5.9, we conclude that ShhA*ii is a symmetric *–k-semiring. h In a similar vein, it is easy to prove the following result for symmetric weak inductive *-semirings. Theorem 5.16. If S is a symmetric weak inductive *-semiring, then so is ShhA*ii. 6. Conclusions This paper is a continuation of our previous work [7]. An equivalent characterization of *–k-semirings is given and several equations as well as inequations which hold in any *–k-semiring are presented. Natural F-preorders on a monoid are introduced and the notion of F-sum-ordered semirings is defined in terms of these preorders. It is proved that the notions of (weak) inductive *-semirings, *–l-semirings and *–k-semirings are equivalent concepts for F-sum-ordered semirings. By virtue of duality, the notions of symmetric *–k-semirings, symmetric *–l-semirings and symmetric weak inductive *-semirings are introduced and some of the relationships among them are investigated. In addition, it is proved that if S is a symmetric *–k-semiring [symmetric weak inductive *-semiring], so is the semiring of formal power series ShhA*ii. Acknowledgements The authors are highly grateful to the referees for their valuable suggestions and detailed comments which greatly helped in improving the paper. References [1] [2] [3] [4] [5] [6] [7]
´ sik, Iteration Theories, Springer, Berlin, 1993. S.L. Bloom, Z. E ´ sik, Matrix and matrical iteration theories, part I, J. Comput. Syst. Sci. 46 (1993) 381–408. S.L. Bloom, Z. E J. Desharnais, B. Mo¨ller, Characterizing determinacy in Kleene algebras, Inform. Sci. 139 (2001) 253–273. ´ sik, W. Kuich, Inductive *-semirings, Theor. Comput. Sci. 324 (2004) 3–33. Z. E ´ sik, H. Leiß, Algebraically complete semirings and Greibach normal form, Ann. Pure Appl. Logic 133 (2005) 173–203. Z. E ´ sik, W. Kuich, Locally closed semirings, Monatsh. Math. 137 (2002) 21–29. Z. E F. Feng, X.Z. Zhao, Y.B. Jun, *–l-semirings and *–k-semirings, Theor. Comput. Sci. 347 (2005) 423–431.
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[8] K. Glazek, A Guide to the Literature on Semirings and their Applications in Mathematics and Information Sciences, Kluwer, Dordrecht, 2002. [9] J.S. Golan, Semirings and Affine Equations over Them: Theory and Applications, Kluwer, Dordrecht, 2003. [10] D. Kozen, On Kleene algebras and closed semirings, in: Proc. MFCS’90, Lecture Notes in Computer Science, vol. 452, Springer, Berlin, 1990, pp. 26–47. [11] W. Kuich, A. Salomaa, Semirings, Automata and Languages, Springer, Berlin, 1986. [12] H. Leiß, Kleene modules and linear languages, J. Logic Algeb. Program. 66 (2006) 185–194. [13] E.G. Manes, M.A. Arbib, Algebraic Approaches to Program Semantics, Springer, Berlin, 1986. [14] A. Tarski, A lattice theoretical fixpoint theorem and its applications, Pacific J. Math. 5 (1955) 285–309.
On *–k-semirings Feng Feng a
a,*
, Young Bae Jun b, Xian Zhong Zhao
c
Department of Applied Mathematics and Applied Physics, Xi’an Institute of Posts and Telecommunications, Xi’an 710061, PR China b Department of Mathematics Education, Gyeongsang National University, Chinju 660-701, South Korea c Department of Mathematics, Northwest University, Xi’an 710069, PR China Received 10 October 2006; received in revised form 11 April 2007; accepted 6 May 2007
Abstract A *–k-semiring is an ordered semiring S equipped with a star operation such that for any a, b 2 S, a*b is the least fixed point of the linear mapping x # ax + b over S. The notion of *–k-semirings is a generalization of several important concepts such as continuous semirings, (weak) inductive *-semirings and the Kleene algebras of Conway and Kozen. We investigate several basic properties of *–k-semirings and obtain results on *–k-semirings in relation to preorders, duality and formal power series. Some of these results can be seen as generalizations of relevant results on inductive *-semirings. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Semiring; *–k-semiring; Preorder; Formal power series
1. Introduction It is well known that fixed points play a fundamental role in computer science. For example, the least and the greatest fixed points of an order preserving function have shown to be the basis of logic of programs. Moreover, context free grammars are indeed systems of fixed point equations over the semiring of languages. All of these facts justify the various attempts devoted to the study of algebraic structures equipped with fixed point operations. On the other hand, semirings have been found useful for solving problems in different areas of applied mathematics and information sciences, since the structure of a semiring provides an algebraic framework for modelling and studying the key factors in these applied areas. The applications of semirings to areas such as optimization theory, graph theory, theory of discrete event dynamical systems, generalized fuzzy computation, automata theory, formal language theory, coding theory and analysis of computer programs have been extensively studied in the literature (cf. [8,9,11]). ´ sik and Kuich [4] defined an inductive *-semiring to be a partially ordered semiring equipped Recently, E with a star operation satisfying the fixed point equation and the fixed point induction rule for linear terms. *
Corresponding author. E-mail addresses: [email protected] (F. Feng), [email protected] (Y.B. Jun), [email protected] (X.Z. Zhao).
0020-0255/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2007.05.019
F. Feng et al. / Information Sciences 177 (2007) 5012–5023
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They showed that any inductive *-semiring is a Conway semiring and an iteration semiring satisfying 1* = 1**. In addition, they proved that both the power series semiring and the matrix semiring of any inductive *-semiring are inductive *-semirings. Following this line of exploration, Feng et al. [7] introduced *–k-semirings and proved that if S is a weak inductive *-semiring, so is the semiring of power series ShhA*ii. This gave a positive answer to one of E´sik and Kuich’s open problems in [4]. The notion of *–k-semirings extends several important concepts such as continuous semirings, (weak) inductive *-semirings and the Kleene algebras [3,10,12] of Conway and Kozen. As a continuation of our previous work [7], we study several basic properties of *–k-semirings in the present paper and obtain results on *–k-semirings relevant to preorders, duality and formal power series. Some of these results can be seen as generalizations of relevant results on (weak) inductive *-semirings. The paper is organized as follows. Section 2 gives some basic concepts and results used for our discussion. In Section 3, we first recall our definitions of *–l-semirings and *–k-semirings which were proposed in [7]. Then we give an equivalent characterization of *–k-semirings and present the basic equations or inequations which hold in any *–k-semiring. In Section 4, we define natural F-preorders on a monoid and study F-sum-preorders on semirings. We introduce the notion of F-sum-ordered semirings which is a straightforward generalization of sum-ordered semirings discussed in [4,13]. We show that the notions of (weak) inductive *-semirings, *–l-semirings and *–k-semirings are equivalent concepts for F-sum-ordered semirings. Section 5 is devoted to duality. We introduce and investigate symmetric *–k-semirings, symmetric *–l-semirings and symmetric weak inductive *-semirings. We discuss some of the relationships among these *-semirings. In addition, we consider the semiring of power series and prove that if S is a symmetric *–k-semiring [symmetric weak inductive *-semiring], so is the semiring of power series ShhA*ii. 2. Basic notions A semiring is an algebra S = (S, +, Æ, 0, 1) equipped with binary operations + (sum or addition) and Æ (product or multiplication) and constants (or nullary operations) 0 and 1 such that the additive reduct (S, +, 0) is a commutative monoid with identity 0, the multiplicative reduct (S, Æ, 1) is a monoid with identity 1 and multiplication distributes over all finite sums, including the empty sum. That is ða þ bÞc ¼ ac þ bc; cða þ bÞ ¼ ca þ cb; a0 ¼ 0a ¼ 0 hold for all a, b, c 2 S. A morphism of semirings is a function that preserves the operations and constants. A semiring S is said to be ordered if it is equipped with a partial order such that the operations on S are monotonic. A morphism of ordered semirings also preserves the partial order. A *-semiring is a semiring S equipped with a star operation * : S ! S. Morphisms of *-semirings also preserve the star operation. An ordered *-semiring means an ordered semiring equipped with a star operation (where the star operation need not be monotonic). A morphism of ordered *-semirings is an order preserving *-semiring morphism. An ordered *-semiring is called an inductive *-semiring [4] if it satisfies the fixed point inequation aa þ 1 6 a
ð1Þ
and the fixed point induction rule ax þ b 6 x ) a b 6 x:
ð2Þ
E´sik and Kuich [4] proved that the following equations: aa þ 1 ¼ a ; a a þ 1 ¼ a ; ðabÞ ¼ 1 þ aðbaÞ b; ðabÞ a ¼ aðbaÞ ; ða þ bÞ ¼ ða bÞ a hold in any inductive *-semiring and the star operation on any inductive *-semiring is monotonic.
ð3Þ ð4Þ ð5Þ ð6Þ ð7Þ
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A weak inductive *-semiring [4] is an ordered *-semiring which satisfies the fixed point equation (3), the sum star equation (7) and the weak fixed point induction rule ax þ b ¼ x ) a b 6 x:
ð8Þ
Let S be a semiring. Then any mapping f of the form x # ax + b, where a, b 2 S, is called a linear mapping over the semiring S. Since both addition and multiplication on an ordered semiring are monotonic, we deduce that any linear mapping over an ordered semiring is monotonic as well. Let S be a poset and let f be a mapping on S. We say p 2 S is a prefixed point of f if f(p) 6 p. The least element of the set of all the prefixed points of f, when it exists, is called the least prefixed point of f and is denoted by l.f. An element p of S is called a fixed point of f if f(p) = p. The least fixed point of f is similarly defined and is denoted by k.f. If f is concretely given by terms, such as f(x) = ax + b, we write the least prefixed [fixed] point of f as lx(ax + b) [kx(ax + b), respectively]. 3. *–l-semirings and *–k-semirings In [7], the authors defined a l-semiring1 [k-semiring] to be an ordered semiring S on which every linear mapping has a least prefixed [fixed] point. They also defined a *–l-semiring [*–k-semiring] to be an ordered *-semiring S such that any linear mapping f : x # ax + b over S has a least prefixed [fixed] point lx.f = a*b [kx.f = a*b, respectively]. Tarski in [14] proved that the least prefixed point of a monotonic function f is a fixed point of f. Thus, we deduce the following. Lemma 3.1 [7]. A l-semiring S is a k-semiring such that kx.f = lx.f for any linear mapping f : x # ax + b over S. In particular, every *–l-semiring is a *–k-semiring satisfying kx:f ¼ lx:f ¼ a b for any linear mapping f : x # ax + b over S. Using Lemma 3.1 and the definitions above, we established in [7] the following result. Proposition 3.2. Suppose that S is an ordered *-semiring. Then the following conditions are equivalent: (1) (2) (3) (4)
S S S S
is is is is
an inductive *-semiring. a *–l-semiring. both a l-semiring and a *–k-semiring. both a l-semiring and a weak inductive *-semiring.
The next result gives an equivalent characterization of *–k-semirings, from which we deduce that any (weak) inductive *-semiring is a *–k-semiring. Lemma 3.3. Let S be an ordered *-semiring. Then S is a *–k-semiring if and only if S satisfies the fixed point equation (3) and the weak fixed point induction rule (8). Hence, S is a *–k-semiring satisfying (7) if and only if S is a weak inductive *-semiring. Proof. Suppose that S is a *–k-semiring. Then for any a, b 2 S, a*b is the least fixed point of the linear mapping x # ax + b over S. In particular, a* is the least fixed point of the mapping x # ax + 1 and so the fixed point equation (3) holds in S. Furthermore, let a, b, x 2 S with ax + b = x. Then x is a fixed point of the linear mapping x # ax + b. But by definition, a*b is the least fixed point of the linear mapping x # ax + b. Hence it follows that a*b 6 x and so S satisfies the weak fixed point induction rule (8).
1
´ sik and Leiß in [5]. Notice that a different notion of l-semirings is given by E
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Conversely, assume that S is an ordered *-semiring satisfying (3) and (8). Then for any linear mapping f : x # ax + b over S, we have f ða bÞ ¼ aða bÞ þ b ¼ ðaa þ 1Þb ¼ a b by the fixed point equation (3). Hence, a*b is a fixed point of f. Moreover, we easily deduce that a*b is the least fixed point of f by the weak fixed point induction rule (8). Thus S is a *–k-semiring as required. h ´ sik and Kuich [4] proved that Eqs. (4)–(6) hold in any inductive *-semiring. However, the following result E says that these equations actually hold in any *–k-semiring (therefore, they also hold in any weak inductive *-semiring). Lemma 3.4. The dual fixed point equation (4), the product star equation (5) and Eq. (6) hold in any *–k-semiring. Proof. Suppose that S is a *–k-semiring. Then by Lemma 3.3, S is an ordered *-semiring satisfying the fixed point equation (3) and the weak fixed point induction rule (8). First, we show that S satisfies the dual fixed point equation (4). In fact, since for any a 2 S aða a þ 1Þ þ 1 ¼ ðaa þ 1Þa þ 1 ¼ a a þ 1; we deduce that a* 6 a*a + 1 by (8). To prove the reverse inequation, notice that for any a, b 2 S,
abaðbaÞ þ a ¼ aðbaðbaÞ þ 1Þ ¼ aðbaÞ : Thus we have (ab)*a 6 a(ba)* by (8). Taking b = 1, we deduce that a*a 6 aa*. Then it follows that a a þ 1 6 aa þ 1 ¼ a ; proving that S satisfies (4). Next, we prove that S satisfies (5). Note first that we have proved above that S satisfies the inequation
ðabÞ a 6 aðbaÞ : Hence, it follows that
aðbaÞ b þ 1 6 abðabÞ þ 1 ¼ ðabÞ : Thus it remains to show the reverse inequation. However, this follows immediately from the fact: abðaðbaÞ b þ 1Þ þ 1 ¼ aðbaðbaÞ þ 1Þb þ 1 ¼ aðbaÞ b þ 1 by the weak fixed point induction rule (8). Finally, using Eqs. (4) and (5) proved above, we deduce
ðabÞ a ¼ ðaðbaÞ b þ 1Þa ¼ aððbaÞ ba þ 1Þ ¼ aðbaÞ and so Eq. (6) also holds in S.
h
By Lemma 3.3, we see that a *–k-semiring S is a weak inductive *-semiring if and only if S satisfies the sum star equation (7). In addition, if S is an ordered *-semiring with a commutative multiplication, we established the following result in [7]. Proposition 3.5. Let S be an ordered *-semiring with a commutative multiplication. Then S is a weak inductive *semiring if and only if S is a *–k-semiring. Proposition 3.6. Any *–k-semiring satisfies the following inequations: 0 6a; a 6a þ b; n X i¼0
ai 6a ; n P 0:
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Proof. Suppose that S is a *–k-semiring. For any a 2 S, since 1 Æ a + 0 = a, we deduce that 0 6 a by (8). Note that the second inequation is equivalent to the first one in any ordered semiring. Thus it holds in S since the first inequation holds as proved above. For the last inequation, we only need to apply the fixed point equation (3) repeatedly and as a result, we have n X a ¼ anþ1 a þ ai i¼0
for any n P 0. Then by the second inequation proved above, it follows that n n X X ai 6 anþ1 a þ ai ¼ a i¼0
i¼0
for any n P 0. h Problem 3.7. Does there exist a *–k-semiring with a noncommutative multiplication which is not a weak inductive *-semiring? Problem 3.8. Does there exist a *–k-semiring with a nonmonotonic star operation? Does there exist a *–ksemiring with a monotonic star operation which is not an inductive *-semiring? 4. F-sum-ordered semirings Recall that a preorder on a set A means a reflexive transitive binary relation on A. Let S be a monoid and F be a submonoid of S. The binary relations 6lF , 6rF and 6jF defined on S by a6lF b () ð9f 2 F Þ fa ¼ b; a6rF b () ð9f 2 F Þ af ¼ b; a6jF b () ð9f1 ; f2 2 F Þ f 1 af2 ¼ b for all a, b 2 S are preorders on S. We refer to 6lF , 6rF and 6jF as left natural F-preorder, right natural F-preorder and two-sided natural F-preorder, respectively. In general, we call them natural F-preorders on S. Natural F-preorders play an important role in diverse areas and especially in semigroup theory. To see this, we would like to point out the following examples. Example 4.1. Let S be a monoid. Then it is easy to see that the equivalence relations 6lS \ ð6lS Þ1 , 6rS \ ð6rS Þ1 and 6jS \ ð6jS Þ1 induced by the natural S-preorders are precisely the Green’s equivalence relations L, R and J on S, respectively. Example 4.2. Let S be an inverse monoid with semilattice of idempotents E. Then the natural partial order on S defined by a 6 b () ð9e 2 EÞ
a ¼ eb
is the converse of the left natural E-preorder 6lE . That is, 6¼ ð6lE Þ1 . Given a semiring S and a subsemiring F of S. Let us consider the additive reduct (S, +, 0) of S. Since F is a subsemiring of S, (F, +, 0) is a commutative submonoid of (S, +, 0). It is clear that the natural (F, +, 0)-preorders on S are all the same, i.e., 6lðF ;þ;0Þ ¼ 6rðF ;þ;0Þ ¼ 6jðF ;þ;0Þ . In this case, we simply denote the natural (F, +, 0)preorder by 6þ F and refer to it as the F-sum-preorder on S. Definition 4.3. Let S be a semiring partially ordered by the relation 6 and F be a subsemiring of S. Then S is called a F-sum-ordered semiring if 6þ F ¼6, i.e., when a 6 b () ð9f 2 F Þ for all a, b 2 S.
b¼f þa
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In particular, if the subsemiring F is the semiring S itself, then the S-sum-ordered semiring coincides with the sum-ordered semiring discussed in [4,13]. Proposition 4.4. Suppose that S is a semiring partially ordered by the relation 6 and F is a subsemiring of S. Then the following conditions are equivalent: (1) 0 is the least element of F, i.e., 0 6 f holds for all f 2 F. (2) For all f, g 2 F, g 6 g + f. (3) 6þ F is a partial order on S included in 6. Proof. It is clear that the first two conditions are equivalent in any ordered semiring. So we only need to show that the first condition implies the third and vice versa. Assume that 0 is the least element of F. If a6þ F b then f + a = b for some f 2 F. Thus it follows that a = 0 + a 6 f + a = b. That is, 6þ is included in 6. Moreover, we easily deduce that F þ 6þ F \ ð6F Þ
1
ð6 \61 Þ ¼ DS ;
þ where DS refers to the identity relation on S. But 6þ F is a preorder, it follows that 6F is a partial order on S included in 6. þ Now suppose that 6þ F is a partial order on S included in 6. For all f 2 F, we have 06F f since 0 + f = f. But þ by assumption, 6F 6. Thus it follows that 0 is the least element of F. h
Proposition 4.5. Let S and S 0 be ordered semirings. Let F be a subsemiring of S and h : S ! S 0 be a semiring morphism. If S is F-sum-ordered and 0 is the least element of h(F), then h is an ordered semiring morphism. Proof. Suppose that a 6 b in S. Then there exists f 2 F such that f + a = b since S is F-sum-ordered. By assumption, h is a semiring morphism and so h(F) is a subsemiring of S 0 , in which we have h(f) + h(a) = h(b). 0 Hence we deduce hðaÞ6þ hðF Þ hðbÞ in S . Moreover, by hypothesis, 0 is the least element of h(F). Thus by Proposition 4.4, it follows that h(a) 6 h(b) and so we conclude that h is an ordered semiring morphism. h The following two results are immediate consequences of the above propositions. Corollary 4.6. Suppose that S is a semiring partially ordered by the relation 6. Then S is a positive semiring, i.e., þ 0 is the least element of S if and only if 6þ S 6 and S is sum-ordered with respect to 6S . Corollary 4.7 [4]. Suppose that S and S 0 are ordered semirings and h : S ! S 0 is a semiring morphism. If S is sum-ordered and 0 is the least element of S 0 , then h is an ordered semiring morphism. To our knowledge, whether the fixed point induction rule (2) and the weak fixed point induction rule (8) are equivalent conditions is an interesting problem which is still open. However, as indicated by the following result, these two conditions are actually equivalent for any F-sum-ordered semiring. Lemma 4.8. Let S be an ordered *-semiring and F be a subsemiring of S. If S is F-sum-ordered, then S satisfies the fixed point induction rule (2) if and only if S satisfies the weak fixed point induction rule (8). Proof. It is clear that if S satisfies the fixed point induction rule (2), then S satisfies the weak fixed point induction rule (8). Thus we only need to show that if S is a F-sum-ordered semiring satisfying (8), then S also satisfies (2). Suppose that a, b, x 2 S with ax + b 6 x. Since S is a F-sum-ordered semiring, there exists f 2 F such that (ax + b) + f = x. Hence, we have a ðb þ f Þ 6 x by (8). But b 6 b + f since S is F-sum-ordered. By monotonicity, it follows that a b 6 a ðb þ f Þ 6 x and we conclude that S satisfies the fixed point induction rule (2). h
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The following result says that the notions of (weak) inductive *-semirings, *–l-semirings and *–k-semirings are equivalent concepts for F-sum-ordered semirings. Theorem 4.9. Suppose that S is an ordered *-semiring and F is a subsemiring of S. If S is F-sum-ordered, then the following conditions are equivalent: (1) (2) (3) (4)
S S S S
is is is is
an inductive *-semiring. a *–l-semiring. a weak inductive *-semiring. a *–k-semiring.
Proof. First, for any ordered *-semiring S (needless to be F-sum-ordered), we have that (1) ) (2) ) (3) ) (4) by Proposition 3.2 and Lemma 3.3. Thus it remains to show that if S is a *–k-semiring which is F-sum-ordered, then S is an inductive *-semiring. So suppose that S is a F-sum-ordered *–k-semiring. Then by Lemma 3.3, we know that S satisfies the fixed point equation (3) and the weak fixed point induction rule (8). Thus it follows that S satisfies the fixed point in Eq. (1). Moreover, by Lemma 4.8, we deduce that S satisfies the fixed point induction rule (2) since it satisfies the weak fixed point induction rule (8). Hence we conclude that S is an inductive *-semiring. h As an immediate consequence derived from the above result, we have Corollary 4.10 [4]. Suppose that S is a *-semiring which is a sum-ordered semiring. Then S is an inductive *-semiring if and only if S satisfies the fixed point (in)equation and the weak fixed point induction rule. Thus, S is an inductive *-semiring if and only if S is a weak inductive *-semiring. 5. Duality E´sik and Kuich [4] introduced the following concepts of duality. Suppose that S is a *-semiring. The dual (or opposite) semiring of S is a *-semiring Sop equipped with the same operations and constants as S except for multiplication, which is the reverse of the multiplication on S. When S is ordered, so is Sop equipped with the same partial order. It is clear that (Sop)op = S. The dual top of a *-semiring term t is defined by induction on the structure of t as follows: w w w w
If If If If
t is a variable or one of the constants 0, 1, then top = t. op t = t1 + t2 then top ¼ top 1 þ t2 . op op op t = t1t2 then t ¼ t2 t1 . t ¼ t1 then top ¼ ðtop 1 Þ .
op Thus by definition, we have (top)op = t. The dual of an equation t1 = t2 is the equation top 1 ¼ t 2 and the dual op op op op of an implication (t1 = s1) ^ ^ (tn = sn) ) t = s is the implication ðt1 ¼ s1 Þ ^ ^ ðtn ¼ sn Þ ) top ¼ sop . The dual of an inequation t 6 s or implication (t1 6 s1) ^ ^ (tn 6 sn) ) t 6 s is defined in the same way. It is easy to see that the dual fixed point equation (4) is indeed the dual of the fixed point equation (3). Similarly, the dual fixed point induction rule
xa þ b 6 x ) ba 6 x
ð9Þ
is the dual of the fixed point induction rule (2). Moreover, an (in)equation (or implication) is said to be self dual if its dual is equivalent to itself. The product star equation is an example of a self dual equation. By definition of duality, it is easy to prove the following. Lemma 5.1 [4]. The dual of an (in) equation or implication holds in a *-semiring S if and only if it holds in Sop. A Conway semiring [1,2,4,6] is a *-semiring S which satisfies the product star equation (5) and the sum star equation (7). By Proposition 36 of [4], the dual of a Conway semiring is a Conway semiring. However, by the following example given by Kozen [10], we claim that the corresponding fact does not hold for *–k-semirings.
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Example 5.2. Let X be an infinite set. Let F be the set of all strict, finitely additive set functions f : PðX Þ ! PðX Þ, i.e., functions f satisfying f ðA [ BÞ ¼ f ðAÞ [ f ðBÞ and f(;) = ;. When f ; g 2 F, define for any A 2 PðX Þ ðf þ gÞðAÞ ¼ f ðAÞ [ gðAÞ; ðf gÞðAÞ ¼ f ðgðAÞÞ: Moreover, let the constants in F be functions 0; 1 2 F such that 0ðAÞ ¼ ;; 1ðAÞ ¼ A for all A 2 PðX Þ. And the star operation on F is given by f ðAÞ ¼ [a f a ðAÞ for all A 2 PðX Þ, where f 0 ðAÞ ¼ A; f aþ1 ðAÞ ¼ f ðf a ðAÞÞ and f k ðAÞ ¼ [a
ð10Þ
Proposition 5.7. A weak inductive *-semiring S is a symmetric weak inductive *-semiring if and only if Sop is a weak inductive *-semiring.
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Proof. Suppose that S is a weak inductive *-semiring. If S is a symmetric weak inductive *-semiring, then S satisfies the dual weak fixed point induction rule (10). Thus by Lemma 5.1, Sop satisfies the weak fixed point induction rule (8). But by Corollary 10 of [4], any weak inductive *-semiring is a Conway semiring. Hence, S is a Conway semiring and it follows that Sop is also a Conway semiring by Proposition 36 of [4]. Thus Sop is a Conway semiring satisfying (8). Moreover, it is easy to verify that in any *-semiring, the product star equation (5) implies the fixed point equation (3). Thus we conclude that Sop is an ordered *-semiring satisfying (3), (7) and (8). Hence, Sop is a weak inductive *-semiring as required. Conversely, if Sop is a weak inductive *-semiring, then S satisfies the dual weak fixed point induction rule (10) by Lemma 5.1. By hypothesis, S is a weak inductive *-semiring. Hence, it follows that S is a symmetric weak inductive *-semiring. h By Lemma 3.3, an ordered *-semiring is a *–k-semiring if and only if it satisfies the fixed point equation (3) and the weak fixed point induction rule (8). Considering duality, we obtain the following result for symmetric *–k-semirings. Lemma 5.8. An ordered *-semiring S is a symmetric *–k-semiring if and only if S satisfies the fixed point equation (3), the weak fixed point induction rule (8) and their duals (4) and (10). Now we establish the next proposition from the above results. Proposition 5.9. Suppose that S is an ordered *-semiring. Then the following conditions are equivalent: (1) S is a symmetric *–k-semiring. (2) S is a *–k-semiring such that Sop is a *–k-semiring. (3) S is a *–k-semiring satisfying the dual weak fixed point induction rule.
Proof. By Lemmas 3.3, 5.1 and 5.8, it is easy to see that the first condition implies the second and the second condition implies the third. It remains to show that the last condition implies the first. So suppose that S is a *– k-semiring satisfying the dual weak fixed point induction rule (10). By Lemma 3.4, we know that S satisfies the dual fixed point equation (4). But by Lemma 3.3, S also satisfies the fixed point equation (3) and the weak fixed point induction rule (8). Thus, it follows that S is a symmetric *–k-semiring by Lemma 5.8. h By Lemma 3.3 and Proposition 5.9, one easily proves that any symmetric weak inductive *-semiring is a symmetric *–k-semiring. Moreover, if the multiplication on S is commutative, we have the following symmetric version of Proposition 3.5. Proposition 5.10. Let S be an ordered *-semiring with a commutative multiplication. Then S is a symmetric *–k-semiring if and only if S is a symmetric weak inductive *-semiring. Proof. As mentioned above, it is easy to prove that if S is a symmetric weak inductive *-semiring, then S is a symmetric *–k-semiring by Lemma 3.3 and Proposition 5.9. It remains to show that if S is a symmetric *–k-semiring with a commutative multiplication, then S is a symmetric weak inductive *-semiring. So suppose that S is a commutative symmetric *–k-semiring. Since S is a symmetric *–k-semiring, we immediately have that S is a *–k-semiring and so S is a weak inductive *-semiring by Proposition 3.5. But by Proposition 5.9, S also satisfies the dual weak fixed point induction rule (10). Hence, we conclude that S is a symmetric weak inductive *-semiring. h By the preceding results, Proposition 3.2 transfers to the symmetric case. Theorem 5.11. Let S be an ordered *-semiring. Then the following conditions are equivalent: (1) (2) (3) (4)
S S S S
is is is is
a a a a
symmetric symmetric symmetric symmetric
inductive *-semiring. *–l-semiring. l-semiring and a symmetric *–k-semiring. l-semiring and a symmetric weak inductive *-semiring.
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Proof. (1. ) 2.) Suppose that S is a symmetric inductive *-semiring. Then S satisfies the fixed point equation (3), the fixed point induction rule (2) and their duals (4) and (9). Thus for any linear mapping f : x # ax + b over S, we have lx:f ¼ a b
ð11Þ
by (2) and (3). Dually, by (4) and (9), we have lx:f op ¼ ba :
ð12Þ
Clearly, it follows from (11) and (12) that S is a symmetric *–l-semiring. (2. ) 3.) Suppose that S is a symmetric *–l-semiring. Then by definition, S is a symmetric l-semiring. For any linear mapping f : x # ax + b over S, by Lemma 3.1, we have kx:f ¼ lx:f ¼ a b and by duality, we have kx:f op ¼ lx:f op ¼ ba : Hence, S is both a symmetric l-semiring and a symmetric *–k-semiring. (3. ) 4.) Assume that S is both a symmetric l-semiring and a symmetric *–k-semiring. We need to show that S is a symmetric weak inductive *-semiring. In fact, since S is a symmetric l-semiring, we know that both lx(ax + b) and lx(xa + b) exist for any a, b 2 S. By assumption, S is also a symmetric *–k-semiring and so we have kx(ax + b) = a*b and kx(xa + b) = ba* for any a, b 2 S. Hence by Lemma 3.1, we deduce that lxðax þ bÞ ¼ kxðax þ bÞ ¼ a b
ð13Þ
and dually we have lxðxa þ bÞ ¼ kxðxa þ bÞ ¼ ba
ð14Þ
for any a, b 2 S. From (13), a* is the least prefixed point of the linear mapping x # ax + 1 and so we deduce that the fixed point inequation (1) holds in S. Similarly, we easily deduce that the fixed point induction rule (2) holds in S by (13). Thus S is an inductive *-semiring and so a weak inductive *-semiring. Furthermore, it follows from (14) that S satisfies the dual (weak) fixed point induction rule. Hence, we conclude that S is both a symmetric lsemiring and a symmetric weak inductive *-semiring. (4. ) 1.) Assume that S is both a symmetric l-semiring and a symmetric weak inductive *-semiring. Then it is clear that S is a weak inductive *-semiring and so S is a *–k-semiring by Lemma 3.3. Since S also satisfies the dual weak fixed point induction rule (10), we deduce that S is a symmetric *–k-semiring by Proposition 5.9. Hence S is both a symmetric l-semiring and a symmetric *–k-semiring. Consequently, as shown in (3. ) 4.), S is an inductive *-semiring satisfying the dual fixed point induction rule (9). Hence, we conclude that S is a symmetric inductive *-semiring. h In the end of this section, we discuss the semirings of formal power series. Let S be a semiring and A be a set. A formal power series over A with coefficients in S is a function r : A ! S; usually denoted by X ðr; uÞu; r¼ u2A
where (r, u) is the value r(u) of function r on the word u, and A* is the free monoid of all words over A including the empty word . Equipped with the operations of pointwise sum and Cauchy product, the set of all formal power series over A with coefficients in S forms a semiring, which is called the formal power series semiring of S and is denote by ShhA*ii. If S is ordered, ShhA*ii is converted into an ordered semiring by the pointwise order. Suppose that S is a *-semiring. For any r 2 ShhA*ii and any u 2 A*, if u = , we define ðr ; Þ ¼ ðr; Þ ;
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otherwise, define
ðr ; uÞ ¼ ðr; Þ
X
ðr; vÞðr ; wÞ:
vw¼u;v6¼
Then we have a star operation on ShhA*ii. ´ sik and Kuich proved that if S is an inductive *-semiring, then so is the formal power series semiring In [4], E * ShhA ii. For *–k-semirings and weak inductive *-semirings, we established in [7] the following results. Theorem 5.12. If S is a *–k-semiring, then so is ShhA*ii. Theorem 5.13. If S is a weak inductive *-semiring, then so is ShhA*ii. To show that the formal power series semiring of any symmetric *–k-semiring is again a symmetric *–ksemiring, we need the following auxiliary result. Proposition 5.14 [4]. Suppose that S is a *-semiring and A is a set. Then the semirings (ShhA*ii)op and SophhA*ii are isomorphic. Theorem 5.15. If S is a symmetric *–k-semiring, then so is ShhA*ii. Proof. Assume that S is a symmetric *–k-semiring. Then by Proposition 5.9, S is a *–k-semiring such that Sop is also a *–k-semiring. Thus by Theorem 5.12, we have that the formal power series semirings ShhA*ii and SophhA*ii are *–k-semirings. But by Proposition 5.14, SophhA*ii is isomorphic to (ShhA*ii)op. Hence, it follows that ShhA*ii is a *–k-semiring such that (ShhA*ii)op is also a *–k-semiring. Thus by Proposition 5.9, we conclude that ShhA*ii is a symmetric *–k-semiring. h In a similar vein, it is easy to prove the following result for symmetric weak inductive *-semirings. Theorem 5.16. If S is a symmetric weak inductive *-semiring, then so is ShhA*ii. 6. Conclusions This paper is a continuation of our previous work [7]. An equivalent characterization of *–k-semirings is given and several equations as well as inequations which hold in any *–k-semiring are presented. Natural F-preorders on a monoid are introduced and the notion of F-sum-ordered semirings is defined in terms of these preorders. It is proved that the notions of (weak) inductive *-semirings, *–l-semirings and *–k-semirings are equivalent concepts for F-sum-ordered semirings. By virtue of duality, the notions of symmetric *–k-semirings, symmetric *–l-semirings and symmetric weak inductive *-semirings are introduced and some of the relationships among them are investigated. In addition, it is proved that if S is a symmetric *–k-semiring [symmetric weak inductive *-semiring], so is the semiring of formal power series ShhA*ii. Acknowledgements The authors are highly grateful to the referees for their valuable suggestions and detailed comments which greatly helped in improving the paper. References [1] [2] [3] [4] [5] [6] [7]
´ sik, Iteration Theories, Springer, Berlin, 1993. S.L. Bloom, Z. E ´ sik, Matrix and matrical iteration theories, part I, J. Comput. Syst. Sci. 46 (1993) 381–408. S.L. Bloom, Z. E J. Desharnais, B. Mo¨ller, Characterizing determinacy in Kleene algebras, Inform. Sci. 139 (2001) 253–273. ´ sik, W. Kuich, Inductive *-semirings, Theor. Comput. Sci. 324 (2004) 3–33. Z. E ´ sik, H. Leiß, Algebraically complete semirings and Greibach normal form, Ann. Pure Appl. Logic 133 (2005) 173–203. Z. E ´ sik, W. Kuich, Locally closed semirings, Monatsh. Math. 137 (2002) 21–29. Z. E F. Feng, X.Z. Zhao, Y.B. Jun, *–l-semirings and *–k-semirings, Theor. Comput. Sci. 347 (2005) 423–431.
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[8] K. Glazek, A Guide to the Literature on Semirings and their Applications in Mathematics and Information Sciences, Kluwer, Dordrecht, 2002. [9] J.S. Golan, Semirings and Affine Equations over Them: Theory and Applications, Kluwer, Dordrecht, 2003. [10] D. Kozen, On Kleene algebras and closed semirings, in: Proc. MFCS’90, Lecture Notes in Computer Science, vol. 452, Springer, Berlin, 1990, pp. 26–47. [11] W. Kuich, A. Salomaa, Semirings, Automata and Languages, Springer, Berlin, 1986. [12] H. Leiß, Kleene modules and linear languages, J. Logic Algeb. Program. 66 (2006) 185–194. [13] E.G. Manes, M.A. Arbib, Algebraic Approaches to Program Semantics, Springer, Berlin, 1986. [14] A. Tarski, A lattice theoretical fixpoint theorem and its applications, Pacific J. Math. 5 (1955) 285–309.