On Representable Mappings of Semigroups into Cardinals

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(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.