Inverse Semigroups Which are Separated Over a Subsemigroup
D. B. McAlister · Transactions of the American Mathematical Society · 1973
An inverse semigroup T is separated over a subsemigroup S if T is generated, as an inverse semigroup, by S and for each $a,b,\epsilon S$ there exists $x\;\epsilon \;Sa \cap Sb$ such that ${a^{ - 1}}a{b^{ - 1}}b = {x^{ - 1}}x$ and dually for right ideals. For example, if T is generated as an inverse semigroup by a semigroup S whose principal left and right ideals form chains under inclusion, then T is separated over S. In this paper we investigate the structure of inverse semigroups T which are separated over subsemigroups S.