On weak regular *-semigroups.
A semigroup \(S\) is called a weak regular *-semigroup if it has a unary operation * satisfying \[ xx^*x=x,\;(x^*)^*=x,\text{ and }(xx^*yy^*)^*=yy^*xx^*\text{ for all }x,y\text{ in }S. \] In this paper a type of partial algebra called a projective partial groupoid is defined. The inverse image of all projections of a fundamental weak regular *-semigroup under the homomorphism induced by the maximum idempotent-separating congruence of a weak regular *-semigroup has a projective partial groupoid structure. Moreover, a weak regular *-product that connects a fundamental weak regular *-semigroup with a corresponding projective partial groupoid is introduced. Every weak regular *-product is a weak regular * -semigroup and any weak regular *-semigroup is constructed in this way.
scientific article; zbMATH DE number 2104916