On congruence lattices of nilsemigroups

Finite Nilsemigroups With Modular Congruence Lattices

“…It gives us the implication from c) to a). The implication from a) to b) is proved in [2]. The rest of the paper is directed to prove that b) leads c).…”

“…N Name Presentation Restrictions 1 A(n) a 2 = ab = ba = b 2 , a n = 0 n 2 2 B 1 (n) a 2 = ab = b 2 , a n = 0 n 3 3 B 2.1 (m, n) a 2 …”

“…Then S has a basis, which consists of maximal elements of S under . This basis form an antichain, so by result of [2], it has 1 or 2 elements. From Theorem 1 we have the following corollary: Table. …”

“…In [2] the characterization of nilsemigroups with distributive and modular congruence lattices had been obtained. The basic notion in that result was the width of a semigroup, considered as a poset under division.…”

“…Recall that the width of a poset is the maximal integer n such that the poset contains an antichain of n elements. It was proved in [2] that the congruence lattice of a nilsemigroup is distributive [modular and not distributive] if and only if it has the width 1 [the width 2].…”