McCarthy, Philip John

English

McCarthy Philip John

University of Szeged

Homomorphisms of certain commutative lattice ordered semigroups By P. J. MCCARTHY in Lawrence (Kansas, U . S. A.)* Let S be a semigroup. It is well-known that a homomorphic image of S is,., to within isomorphism, of the form S/6 where 6 is a congruence on S. In this note-we shall show that if S is a commutative lattice ordered semigroup [2, Chapter XII],. with certain additional properties, then only congruences of a certain type are-required to describe all of the homomorphic images of S. Then we shall point out a particularly interesting example of a class of semigroups which have all of these-properties. In this note, when we refer to a homomorphism from one partially ordered semigroup into another we shall always mean one that preserves ordering. First, assume that S is a commutative partially ordered semigroup, so that there is a partial ordering on S with the property that if a, b, c£ S and if a^b then. ac S be. Also assume that S has an identity e such that a S e for all a£S. Let M be a subsemigroup of S. For each a£S we set a' — {x £ S\mx ^ a for some m£M). Since m^e for all m^M we have maSa and so a£a' for all a£S. We define a relation 9 on S by a = b(6) if and only if a' = b'. This is an equivalence-relation on S and we easily verify the following facts: (1) if a£b' then a'^b', (2) if a = b(6) then ac = bc(6) for all c£S, (3); if a^c^b and a = b(6) then a = c{d). Thus, 0 is a congruence on S and we can consider the semigroup S/0. We denote-the equivalence class of a 6 S with respect to 9 by 9(a), and we also denote the natural, homomorphism from S onto S/0 by 9. From now on we shall assume that S is a lattice with respect to its partial ordering and that a(b v c) = abv ac for all a, b, c£S. (4) if a = b(9) we have a v c = 6 v c ( 0 ) and a A c = 6 A c(d) for al I c 6 S. For, since a£b' there is an m^M such that ma^b. Then m(a v c) = mav mc ^ S B V C and M(AAC)^MA/\MC-^BAC. Hence a v c C ( A v c ) ' and AAC£(BAC)', and so ( a v c ) ' £ ( i v c ) ' and (a A C)' £ (b A C)'. By symmetry, (bv c)' Q(AV C)' and. 0 & A c ) ' g ( a A c ) ' . *) Supported in part by N S F GP-1738. '64 P. J. McCarthy We can therefore define operations v and A on S/0 by 0(a) v 0(b) = 0(a v b) and 0(a) A 6(b) — 6(a A b), and with respect to these operations, 5/0 is a lattice ordered semigroup. It is quite trivial that the properties required of the operations v and A hold. It remains to verify that the ordering induced on 5/0 by the lattice structure is compatible with the multiplication on 5/0. We have 0(a) S 0(6) if and only if •6(b) = 0(a) v 0(b) = 0(avb), and so if 0(a) 0(b) we have for all 0(c) €5/0, 0(bc) = = 6(b)6(c) = 6(a v b)O(c) = 0((a v b)c) = 6(ac v be). Hence 6(a)0(c) S 0(b)0(c). Note that -6(e) is the identity of 5/0 and that 0(a) S 0(e) for all 0(a) € 5/0. Now consider a homomorphism h from 5 onto a partially ordered semigroup T. If'we set M = {m £ 5|/z (m) = h (e)} then M is a subsemigroup of 5. Let 0 be the congruence on 5 associated with M in the manner we have described. If a£ 5 we set fO(a) = h(a). If we show that / is a well-defined mapping from 5/0 into T, then it is clear that / i s a homomorphism from 5/0 onto T such that fO=h. Suppose that 0(a) = 0(b), i.e., a = b(6). Then there are elements m, n£M such that ma^b and nb^a. Hence nma^nb^a and so h(a) = h(mna)^h(nb)=h(b)^h(a). Thus h(a)=h(b) and we conclude that / is well-defined. We seek conditions under which / will bean isomorphism. A suitable condition, for our purposes, is that both 5 and T be residuated [2, p. 189] and that h preserve residuals. For, suppose that this is the case, and that h(a) =h(b). Then h(e)h(b) = h(a) and so h(e)^h(a):h(b). Hence h(e) = h(a):h(b)=h(a:b), which means that a\b£M. Since (a:b)b^a we have b'Qa'. Similarly a'^b' and therefore a=b(6) and 0(a) = — 0(b). We can summarize all of this as the T h e o r e m . Let 5 be a commutative residuated lattice ordered semigroup with an identity e such that aSe for all a£ S. Let T be a residuated partially ordered semi-group and suppose there is a homomorphism h from S onto T which preserves residuals. Then there is a subsemigroup M of 5 such that if 0 is the congruence on 5 determined .as above by M, then there is an isomorphism f from SjO onto T such that fO = h. R e m a r k 1. If 5 and Ta re as in the statement of the theorem, then T becomes a lattice ordered semigroup when we define meet and join on T by h(a) A h(b) = = h(a A b) and h(a) v h(b) = h(a v b). Re mark 2. Let 5 be as in the statement of the theorem, let M be a subsemigroup of 5, and let 0 be the congruence on 5 determined by M. Then the semigroup 5/0 is residuated and the homomorphism 0 preserves residuals. To show this we shall verify that 0(a :b) is the residual of 6(a) by 0(b). We have 0(a: b)O(b) = 0((a: b)b) S 0(a). Furthermore, suppose that 0(c)6(b)^0(a). Then 0(cb)^0(a) and so 0(cb) = = 0(a) A 0(cb) = 0(a A cb). Thus, for some m € M, mcb ^aAcb^a. Hence mc^a:b and so 0(mc)^0(a:b). Since 0(m) = 6(e), as is easily seen, 0(mc) — 0(m)0(c) = 0(c). Hence 0(c)^0(a:b). Therefore, 0(a:b) is the required residual [2, p. 189]. More generally, these conditions are satisfied by the residuated multiplicative lattices, which have been studied by WARD a n d D i L W O R T H (see [1] and the references at the end of that paper). Let L be a residuated multiplicative lattice: then L is a commutative residuated lattice ordered semigroup with an identity / such that A^I for all A £L. The for-mation of the multiplicative lattice L/0, where 0 is determined as above by a sub-semigroup (i.e., multiplicatively closed set) of L, is an abstract construction of the Homomorphisms of lattice ordered semigroups 65 lattice of ideals of a ring of quotients of a commutative ring with identity. A special case of this construction was discussed by DILWORTH [ 1 , pp. 4 8 9 — 4 9 1 ] . Let L be a Noether lattice and let D£L. If M is the set of all A £ L such that D is not greater than or equal to any of the primes associated with a normal decomposition of A, then M is a subsemigroup of L, and if 0 is the congruence on L determined as above by M, then Lj9 is precisely the congruence lattice LD of Dilworth. In particularl if D is a prime P of L then A £ M if and only if A $ P, for if A S P then some minima, prime associated with a normal decomposition of A must be less than or equal to P. References [1] R. P. DILWORTH, Abstract commutative ideal theory, Pacific J. Math., 12 (1962), 481—498. [2] L. FUCHS, Partially Ordered Algebraic Systems (Reading, Mass., 1963). (Received April 15, 1965) 5 A 

Homomorphisms of certain commutative lattice ordered semigroups

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Homomorphisms of certain commutative lattice ordered semigroups

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