Semigroups with complemented congruence lattices

Semigroups with atomistic congruence lattices

“…A straightforward verification shows that S is a semigroup. This construction also appears in [1] and [2], a similar one in [4]. By the latter paper it follows that each ideal of such a semigroup is a retract.…”

Weakly reductive semigroups with atomistic congruence lattices

“…A straightforward verification shows that S is a semigroup. Similar constructions appear in [1], [2], [7], [9], [12].…”

“…implies that a = sa t _, t for some s, t € S l and thus a = sa t _ 1 ] is a finite c h a i n w h e n e v e r h{J y , J x ) < n . L e t u = a o p l a l ---p n a n = v for certain p { e A t 5 such that u Ja, v Jb and a, e J(a) for all / .…”

“…Here it may happen that the locally finite tree X of S = {X; I a , f a ") has no least element and so in (ii) the condition "for x e /* and y e I* p there exists y <a, fi such that xf a = yf p " must be added. An example in [1] shows that this is really necessary. In [2, Section 5] a necessary and sufficient condition for this property is given.…”

Generalised retract semigroups

“…Furthermore, if X has a least element /x then by definition I* is closed under multiplication and thus it is a simple semigroup. Tully semigroups appear in different contexts in semigroup literature (see [1,2] and [3]).…”