Construction of regular semigroups with inverse transversals

Abstract

Let S be a regular semigroup. An inverse subsemigroup S ° of S is an inverse transversal if | V ( x )∩ S °| = 1 for each x ∈ S , where V ( x ) denotes the set of inverses of x . In this case, the unique element of V ( x )∩ S ° is denoted by x °, and x °° denotes ( x °) –1 . Throughout this paper S denotes a regular semigroup with an inverse transversal S °, and E ( S °) = E ° denotes the semilattice of idempotents of S °. The sets { e ∈ S : ee ° = e } and { f ∈ S : f ° f = f } are denoted by I s and Λ s , respectively, or simply I and Λ. Though each element of these sets is idempotent, they are not necessarily sub-bands of S . When both I and Λ are sub-bands of S , S ° is called an S-inverse transversal . An inverse transversal S ° is multiplicative if x ° xyy °∈ E °, and S ° is weakly multiplicative if ( x ° xyy °)°∈ E ° for every x, y ∈ S . A band B is left [resp. right ] regular if e f e = e f [resp. e f e = f e ], and B is left [resp. right ] normal if e f g = e g f [resp. e f g = f e g ] for every e, f, g ∈ B . A subset Q of S is a quasi-ideal of S if QSQ ⊆ S .