Recent developments in inverse semigroup theory

An algebraic characterisation of ample type I groupoids

“…For inverse semigroups the standard reference is [12]. The survey article [16] describes recent advances and the state of the art in inverse semigroup theory. Further [15] and [16,Sections 3 and 4] provide an introduction to Boolean inverse semigroups.…”

“…The survey article [16] describes recent advances and the state of the art in inverse semigroup theory. Further [15] and [16,Sections 3 and 4] provide an introduction to Boolean inverse semigroups. It has to be pointed out that the definition of Boolean inverse semigroups used in the literature has changed over the years, so that care is due when consulting older material.…”

“…It follows from [14, Duality Theorem] that these two operations are dual to each other. See also [16,Theorem 4.4] and [19,Theorem 4.2]. We will not need to specify the morphisms of the categories involved in this duality.…”

“…Since the discovery of noncommutative Stone duality an important aspect of this theory was to establish a dictionary between properties of ample groupoids and properties of Boolean inverse semigroups. A summary of the results obtained so far can be found in Lawson's survey article [16]. From an operator algebraic and representation theoretic point of view one fundamental side of groupoids and semigroups is the property of being CCR and the property of being type I.…”

Inverse Semihypergroups

“…In this paper, we try, based on the paper of Lawson on inverse semigroups [17], to define inverse semihypergroups and prove some basic properties of inverse semihypergroups in analogy with those of inverse semigroups. We include several examples and counter-examples that demonstrate the complexity of the theory we are creating.…”