Weakly left ample semigroups form a class of semigroups that are (2; 1)-subalgebras of semigroups of partial transformations, where the unary operation takes a transformation to the identity map in the domain of. They are semigroups with commuting idempotents such that the e R-class of any element a contains a (necessarily unique) idempotent a + , e R is a left congruence, and the ample identit...

We study the algebraic structure of the semigroup of all 2 × 2 tropical matrices under multiplication. Using ideas from tropical geometry, we give a complete description of Green’s relations and the idempotents and maximal subgroups of this semigroup.

We describe decompositions of the group of units of a ring and of its subgroups, induced by idempotents with certain properties. The results apply to several classes of rings, most notably to semiperfect rings.

A restriction category is an abstract formulation for a category of partial maps, defined in terms of certain specified idempotents called the restriction idempotents. All categories of partial maps are restriction categories; conversely, a restriction category is a category of partial maps if and only if the restriction idempotents split. Restriction categories facilitate reasoning about parti...