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A note on the lattice of weak L-valued congruences on the group of prime order
Fuzzy Sets and Systems 22 (1987) 321-324 North-Holland
321
SHORT COMMUNICATION
A NOTE ON THE LATFICE OF WEAK L-VALUED CONGRUENCES ON THE GROUP OF PRIME ORDER B. ~E~ELIA and G. V O J V O D I ~ Novi Sad, Yugoslavia Received December 1985 Revised May 1986 Weak L-valued (fuzzy) congruences on an algebra, where L is a lattice, were induced in [4]. In this note, we give a representation theorem for one subdirect power of L, by means of the lattice of L-valued congruences on any cyclic group of prime order.
Keywords: Fuzzy relations, Lattices.
Let ~t = (A,.,-1, e) be a group, and let (L, A, V, 0, 1) be a lattice (in the following denoted by L) with 0 and 1 (complete, if A is infinite). We consider L-valued (fuzzy) sets on A as the mappings X:A---> L. The (ordinary) subsets of A (including A and 0) are identified with their characteristic function (0, 1 e L). 1. A weak L-valued congruence relation on ~/[4] is a mapping 15:A2--~ L, such that (i) /~(e, e) = 1 (weak reflexivity) 1, (ii) t)(x, y) =/5(y, x) (symmetry), (iii) t3(x, y)/>/)(x, z) A ~(Z, y) (transitivity), (iv) /~(x. u, y- v) >t/~(x, y) A /3(U, O) and f)(x -1, y-l)/>/)(x, y) (substitution property). The set of all weak L-valued congruences on ~¢ is denoted by Cw(~t). It was proved in [4] that (Cw(s¢), ~<) (where ~t is any algebra, and L is complete) is a complete lattice (as usual, for/3, O e Cw(~t), ~ ~< O if and only if t)(x, y) ~< (r(x, y) for all x, y e A). 2. If L is any lattice, then
(Lr, A, V),
where
Lr = {(P, q) eL2 IP <~q} l In general, if ~¢ = (A, F ) is an algebra and K is the set of its constants, then we require that /)(c, c) = 1, for every c e K [4].
0165-0114/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)
B. Segelja, G. Vojvodi~
322
and the operations are defined componentwise, is obviously a sublattice of L z, namely, one subdirect power of L (denoted by L~). Theorem. Let ~ = ( A , . , - a , e) be a finite group of prime order, and L an arbitrary lattice with 0 and 1. Then: (a) ~ :A2---~L is a weak L-valued congruence on sO, if and only if there are p, q • L, p >- q, such that fg(e, e) = 1, fa(x, x) = p for all x ~ A \ {e}, and f)(x, y) = q otherwise. (b) The lattice (Cw(~¢), ~<) of all weak L-valued congruences on ~¢ is isomorphic to Lr. Proof. (a) Necessity: It is obvious that ~(e, e ) = 1. To prove that ~(x, x) is constant on A \ {e}, there is no harm in assuming that ~(a, a) ~ ~(b, b), for some a, b • A \ { e } , a ~: b. Then put Oa = {x • A I/3(x, x) I> O(a, a)}. Da is a subgroup of ~¢. Indeed, if x, y • Da, then
fig(xy-1, xy-1)
>>.
~(x-1, y - l )
^/3(y, y) = 13(x, x) ^/3(y, y) I> 13(a, a)
(i.e. xy -1 • Da). Now, a • Da, but b ~ D~, since t~(b, b) ;~ ~(a, a), by assumption. This is a contradiction, since ~d has no proper subgroups. Thus, ~(x, x) is constant on A \ {e}, i.e. ~(x, x) = p for all x • A \ {e}. By transitivity we have ~(x, x) t> 13(x, y) ^ p(y, x) =/)(x, y), for all x, y • A. In particular, p I> l)(x, y), for all x, y • A \ {e}. For a, b • A, a #: b, let
Pab = {(X, y) e A 21 p(X, y) >>-~(a, b)}. We show that P~b is a congruence on ~/. Indeed, for x • A, /5(x, x) ~>p/>/5(a, b), and so (x, x) • P,b. The relation P~b is clearly symmetric. It is also transitive: for (x, y), (y, z) • P~b we have /3(x, z) t>/3(x, y) ^/3(y, z) I> 13(a, b) ^/3(a, b) =/)(a, b). Moreover, for (x, y) • P~b and c • A, we have
f)(cx, cy) >- f)(c, c) A fa(X, y) >-p ^ f~(a, b) >- fg(a, b). It is easy to see that (x, y) • Pab implies (x -a, y-a) • P,b. Thus, P~b is a congruence on ~¢. This congruence determines a normal subgroup G of ~/: G = ({x • A I(e, x) • PaO}, ")-
By assumption, G = ( { e } , - ) or G = M , and hence P o b = { ( x , x ) [ x • A } , or P,b = A 2. Since (a, b) • Pab, and a ~ b, it follows that P.b = A 2. Thus, t3(x, y) 1>13 (a, b) for all x, y • A. Since a, b (a :/: b) are arbitrary elements of A, it follows
323
Weak L-valued congruences
that f3 is constant on B =A2\{(x, x) lx cA}. Denoting by q the value of/5 on B, we get that p I> q, which concludes the proof of necessity. Sufficiency: Direct verification. (b) This statement is now obvious. [] Remark. It is clear that a weak L-valued congruence do not reduce to the usual one if L = {0, 1}. In fact, every weak L-valued congruence /~ on an arbitrary group G determines one L-valued subgroup H of G (see [4]): for x • G, H(x) =/5(x, x) (L-valued subgroups were considered in [3]). Hence, if L = {0, 1}, H is an ordinary subgroup of G. L-valued congruences (not weak) with usual reflexivity (~(x, x ) = 1, for all x • G) were defined in [5], and for L = {0, 1} they reduce to the usual ones. The corresponding characterization theorem (in the case of the group of prime order) is: (a)/)(x, x) = 1, for all x, and (b) the lattice of all such relations is isomorphic to L. Example. The lattice (Cw(,d), ~ ) of weak L-valued congruences on the group of prime order, where L is a k-chain (k e N), is given here by its Hasse diagram (Figure 1).
K-1 IK-2
,
,1
Fig. 1. Left: L; right: (Cw(M), ~<).
K
\\
B. Segel]a, G. Vo]vodi(
324
For k = 3 (i.e. when L = {0, p, 1}) and ,~ = ((e, a}, -), this lattice and the list of all weak L-valued congruences on ~ are the following (Figure 2): ~ea a
e a
eJlO al00
ell0 a 10p
eJ~pa p
~ ~lTea 11 11
i Pll
Fig. 2. As Figure 1, for k = 3 .
References [1] J.M. Anthony and H. Sherwood, A characterization of Fuzzy Subgroups, Fuzzy Sets and Systems 7 (1982) 297-305. [2] G. Birkhoff, Lattice Theory, 3rd edition, Colloq. Publ. Vol. 25 (Amer. Math. Soc., Providence, RI, 1967). [3] M. Delorme, Sous-groupes fious, Seminaire 'Mathematique Floue', Lyon (1978-79). [4] G. Vojvodi6 and B. Segelja, On the lattice of weak fuzzy congruence relations on algebras, Review of Research, Fac. of Science, Novi Sad 15(1) (1985) 199-207. [5] G. Vojvodi6 and B. Segeija, On fuzzy quotient algebras, Review of Research, Fac. of Science, Univ. of Novi Sad 13 (1983) 279-288.
321
SHORT COMMUNICATION
A NOTE ON THE LATFICE OF WEAK L-VALUED CONGRUENCES ON THE GROUP OF PRIME ORDER B. ~E~ELIA and G. V O J V O D I ~ Novi Sad, Yugoslavia Received December 1985 Revised May 1986 Weak L-valued (fuzzy) congruences on an algebra, where L is a lattice, were induced in [4]. In this note, we give a representation theorem for one subdirect power of L, by means of the lattice of L-valued congruences on any cyclic group of prime order.
Keywords: Fuzzy relations, Lattices.
Let ~t = (A,.,-1, e) be a group, and let (L, A, V, 0, 1) be a lattice (in the following denoted by L) with 0 and 1 (complete, if A is infinite). We consider L-valued (fuzzy) sets on A as the mappings X:A---> L. The (ordinary) subsets of A (including A and 0) are identified with their characteristic function (0, 1 e L). 1. A weak L-valued congruence relation on ~/[4] is a mapping 15:A2--~ L, such that (i) /~(e, e) = 1 (weak reflexivity) 1, (ii) t)(x, y) =/5(y, x) (symmetry), (iii) t3(x, y)/>/)(x, z) A ~(Z, y) (transitivity), (iv) /~(x. u, y- v) >t/~(x, y) A /3(U, O) and f)(x -1, y-l)/>/)(x, y) (substitution property). The set of all weak L-valued congruences on ~¢ is denoted by Cw(~t). It was proved in [4] that (Cw(s¢), ~<) (where ~t is any algebra, and L is complete) is a complete lattice (as usual, for/3, O e Cw(~t), ~ ~< O if and only if t)(x, y) ~< (r(x, y) for all x, y e A). 2. If L is any lattice, then
(Lr, A, V),
where
Lr = {(P, q) eL2 IP <~q} l In general, if ~¢ = (A, F ) is an algebra and K is the set of its constants, then we require that /)(c, c) = 1, for every c e K [4].
0165-0114/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)
B. Segelja, G. Vojvodi~
322
and the operations are defined componentwise, is obviously a sublattice of L z, namely, one subdirect power of L (denoted by L~). Theorem. Let ~ = ( A , . , - a , e) be a finite group of prime order, and L an arbitrary lattice with 0 and 1. Then: (a) ~ :A2---~L is a weak L-valued congruence on sO, if and only if there are p, q • L, p >- q, such that fg(e, e) = 1, fa(x, x) = p for all x ~ A \ {e}, and f)(x, y) = q otherwise. (b) The lattice (Cw(~¢), ~<) of all weak L-valued congruences on ~¢ is isomorphic to Lr. Proof. (a) Necessity: It is obvious that ~(e, e ) = 1. To prove that ~(x, x) is constant on A \ {e}, there is no harm in assuming that ~(a, a) ~ ~(b, b), for some a, b • A \ { e } , a ~: b. Then put Oa = {x • A I/3(x, x) I> O(a, a)}. Da is a subgroup of ~¢. Indeed, if x, y • Da, then
fig(xy-1, xy-1)
>>.
~(x-1, y - l )
^/3(y, y) = 13(x, x) ^/3(y, y) I> 13(a, a)
(i.e. xy -1 • Da). Now, a • Da, but b ~ D~, since t~(b, b) ;~ ~(a, a), by assumption. This is a contradiction, since ~d has no proper subgroups. Thus, ~(x, x) is constant on A \ {e}, i.e. ~(x, x) = p for all x • A \ {e}. By transitivity we have ~(x, x) t> 13(x, y) ^ p(y, x) =/)(x, y), for all x, y • A. In particular, p I> l)(x, y), for all x, y • A \ {e}. For a, b • A, a #: b, let
Pab = {(X, y) e A 21 p(X, y) >>-~(a, b)}. We show that P~b is a congruence on ~/. Indeed, for x • A, /5(x, x) ~>p/>/5(a, b), and so (x, x) • P,b. The relation P~b is clearly symmetric. It is also transitive: for (x, y), (y, z) • P~b we have /3(x, z) t>/3(x, y) ^/3(y, z) I> 13(a, b) ^/3(a, b) =/)(a, b). Moreover, for (x, y) • P~b and c • A, we have
f)(cx, cy) >- f)(c, c) A fa(X, y) >-p ^ f~(a, b) >- fg(a, b). It is easy to see that (x, y) • Pab implies (x -a, y-a) • P,b. Thus, P~b is a congruence on ~¢. This congruence determines a normal subgroup G of ~/: G = ({x • A I(e, x) • PaO}, ")-
By assumption, G = ( { e } , - ) or G = M , and hence P o b = { ( x , x ) [ x • A } , or P,b = A 2. Since (a, b) • Pab, and a ~ b, it follows that P.b = A 2. Thus, t3(x, y) 1>13 (a, b) for all x, y • A. Since a, b (a :/: b) are arbitrary elements of A, it follows
323
Weak L-valued congruences
that f3 is constant on B =A2\{(x, x) lx cA}. Denoting by q the value of/5 on B, we get that p I> q, which concludes the proof of necessity. Sufficiency: Direct verification. (b) This statement is now obvious. [] Remark. It is clear that a weak L-valued congruence do not reduce to the usual one if L = {0, 1}. In fact, every weak L-valued congruence /~ on an arbitrary group G determines one L-valued subgroup H of G (see [4]): for x • G, H(x) =/5(x, x) (L-valued subgroups were considered in [3]). Hence, if L = {0, 1}, H is an ordinary subgroup of G. L-valued congruences (not weak) with usual reflexivity (~(x, x ) = 1, for all x • G) were defined in [5], and for L = {0, 1} they reduce to the usual ones. The corresponding characterization theorem (in the case of the group of prime order) is: (a)/)(x, x) = 1, for all x, and (b) the lattice of all such relations is isomorphic to L. Example. The lattice (Cw(,d), ~ ) of weak L-valued congruences on the group of prime order, where L is a k-chain (k e N), is given here by its Hasse diagram (Figure 1).
K-1 IK-2
,
,1
Fig. 1. Left: L; right: (Cw(M), ~<).
K
\\
B. Segel]a, G. Vo]vodi(
324
For k = 3 (i.e. when L = {0, p, 1}) and ,~ = ((e, a}, -), this lattice and the list of all weak L-valued congruences on ~ are the following (Figure 2): ~ea a
e a
eJlO al00
ell0 a 10p
eJ~pa p
~ ~lTea 11 11
i Pll
Fig. 2. As Figure 1, for k = 3 .
References [1] J.M. Anthony and H. Sherwood, A characterization of Fuzzy Subgroups, Fuzzy Sets and Systems 7 (1982) 297-305. [2] G. Birkhoff, Lattice Theory, 3rd edition, Colloq. Publ. Vol. 25 (Amer. Math. Soc., Providence, RI, 1967). [3] M. Delorme, Sous-groupes fious, Seminaire 'Mathematique Floue', Lyon (1978-79). [4] G. Vojvodi6 and B. Segelja, On the lattice of weak fuzzy congruence relations on algebras, Review of Research, Fac. of Science, Novi Sad 15(1) (1985) 199-207. [5] G. Vojvodi6 and B. Segeija, On fuzzy quotient algebras, Review of Research, Fac. of Science, Univ. of Novi Sad 13 (1983) 279-288.