On representing semigroups with subsemilattices. (Q360174)

An assertion semigroup (or \(A\)-semigroup) is a structure \((\mathbf S,E)\) where \(\mathbf S\) is a semigroup and \(E\) is a distinguished subsemilattice of \(\mathbf S\). \(A\)-semigroup \(\mathbf S\) is called weakly representable as an \(A\)-semigroup of binary relations if there is a faithful semigroup representation \(\varphi\colon\mathbf S\to\mathcal B(X)\), where \(\mathcal B(X)\) is the semigroup of all binary relations on a set \(X\), such that the elements of \(E\) are represented as restrictions of the identity map. The representation \(\varphi\) is a strong representation if the elements of \(E\) are the only elements of \(\mathbf S\) represented as restrictions of the identity map. Along with the semigroup \(\mathcal B(X)\) the semigroups \(\mathcal P(X)\) and \(\mathcal T(X)\) are considered which are the semigroups of all partial transformations and of all partial one-to-one transformations on \(X\), respectively. The corresponding classes of representable \(A\)-semigroups are quasi-varieties but it is shown that they cannot be finitely axiomatized in first order logic. The method is used to establish the absence of a finite axiomatisation in various classes of semigroups, unary semigroups and ordered semigroups. Along with new results new proofs of some known facts concerning non-finite axiomatisability are presented.