An alternative look at the structure of graph inverse semigroups

Abstract

For any graph inverse semigroup G(E) we describe subsemigroups D 0 = D ∪ {0} and J 0 = J ∪ {0} of G(E) where D and J are arbitrary D-class and J -class of G(E), respectively. In particular, we prove that for each D-class D of a graph inverse semigroup over an acyclic graph the semigroup D 0 is isomorphic to a semigroup of matrix units. Also we show that for any elements a, b of a graph inverse semigroup G(E), J a · J b ∪ J b · J a ⊂ J 0 b if there exists a path w such that s(w) ∈ J a and r(w) ∈ J b .

Embedding of graph inverse semigroups into CLP-compact topological semigroups

“…Graph inverse semigroups play an important role in the study of rings and C * -algebras (see [1,4,17,28,35]). Algebraic theory of graph inverse semigroups is well developed (see [2,8,10,26,27,30,32]). Topological properties of graph inverse semigroups were investigated in [9,11,12,13,33].…”

“…Consider case (ii). By [8,Corollary 5], D e ∪ {0} is a subsemigroup of G(E) which is isomorphic to the semigroup of I e ×I e -matrix units which contradicts condition (4).…”