Equational bases for some 0-direct unions of semigroups (Q2754985)

An involution semigroup is an algebra \({\mathcal I}=(S,\cdot,{}^*)\) such that \((S,\cdot)\) is a semigroup and \(^*\) is a unary operation on \(S\) such that the identities \((xy)^*=y^*x^*\) , \((x^*)^*=x\) hold in \(S\). In this paper the authors investigate the identities satisfied by \(0\)-direct unions of a semigroup with its anti-isomorphic copy, which serve as the standard tool for showing that an arbitrary semigroup can be embedded in an involution semigroup. Given the set of semigroup identities they satisfy, the involution defined on such \(0\)-direct unions can be captured by only two additional identities involving the unary operation symbol. As a corollary of a result on finiteness of equational bases for such involution semigroups, the authors present an involution semigroup consisting of \(13\) elements and not having a finite equational basis.