Embedding of a restriction semigroup into a \(W\)-product. (Q467532)

Restriction (also called ample) semigroups are considered as non-regular generalizations of inverse semigroups; they are algebras of type \((2,1,1)\) where both unary operations assign an idempotent (one-sided identity satisfying certain conditions) to each element. Here is provided a necessary and sufficient condition for a restriction semigroup to be embeddable in a \(W\)-product of a semilattice by a monoid; the condition involves constructing a join of transitive closures of recursively defined binary relations. The author proved earlier that each restriction semigroup has a proper cover embeddable in a \(W\)-product of a semilattice by a monoid.