Proper covers of restriction semigroups and \(W\)-products. (Q2888841)

An algebra \((S,\cdot,+,*)\) of type \((2,1,1)\) is called a restriction semigroup if \((S,\cdot)\) is a semigroup, \((S,\cdot,+)\) satisfies the identities \(x^+x=x\), \(x^+y^+=y^+x^+\), \((x^+y^+)^+=x^+y^+\), \(xy^+=(xy)^+x\), \((S,\cdot,*)\) satisfies the dual identities, and \((x^+)^*=x^+\), \((x^*)^+=x^*\).NEWLINENEWLINE The family of restriction semigroups is considered which consists of the factor semigroups of the free restriction semigroups over congruences contained in the least cancellative congruence. It is proved that every semigroup from this family is embeddable into a \(W\)-product of a semilattice by a monoid. Consequently, it is established that each restriction semigroup has a proper (ample) cover embeddable into such a \(W\)-product.