Equational bases for some 0-direct unions of semigroups

In this paper, we investigate 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 (a semigroup reduct of) an involution semigroup. We show that, 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, we present an involution semigroup (which is, however, not an inverse one) consisting of 13 elements and not having a finite equational basis.