@MISC{_onrepresentable, author = {}, title = {ON REPRESENTABLE MAPPINGS OF SEMIGROUPS INTO CARDINALS}, year = {} }

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Abstract

Let S be a semigroup and f be a mapping of S into the class of nonzero cardinal numbers. The mapping f is said to be representable if there exist a groupoid G and a homomorphism h of G onto S such that Ker(h) is the least congruence of G for which the corresponding factor is a semigroup and f(a) = Cardh −1 (a) for all a ∈ S. The investigation of representable mappings (see the papers [2] and [3]) is closely connected with and originates from the study of the notion of associativity semihypergroup, which was introduced in [4] and further studied e.g. in [1]. The purpose of the present paper is to introduce a new condition (C), necessary for the representability of a mapping f on a semigroup S; our condition, which is a refinement of a similar condition from [3], turns out to be also sufficient on a large class of semigroups. The necessity and the (restricted) sufficiency of (C) will be the two main results of this paper. In their proofs we shall make use of the following rather simple observation of set-theoretical character, which we are not going to prove. LEMMA. Let A be a nonempty set and K be a system of pairwise disjoint nonempty sets. The following two conditions are equivalent: (1) there exists a mapping g of ⋃ K onto A such that A × A is the only equivalence on A containing all the relations g(K)×g(K) (K ∈ K); (2) CardA+CardK ≤ 1+Card ⋃ K. For a semigroup S and an element a ∈ S we denote by Ma the set of the pairs (b,c) of elements of S such that bc = a; denote by Ea the equivalence on Ma generated by the pairs ((uv,w),(u,vw)) where u,v,w are elements with uvw = a; and denote by νa the number of the blocks of Ea. (If Ma is empty then νa = 0.) We introduce the following condition for a mapping f of S into the class of nonzero cardinals: (C) f(a)+νa ≤ 1+ ∑