Superextensions of three-element semigroups

Abstract

A family $\mathcal{A}$ of non-empty subsets of a set $X$ is called an upfamily if for each set $A\in\mathcal{A}$ any set $B\supset A$ belongs to $\mathcal{A}$. An upfamily $\mathcal L$ of subsets of $X$ is said to be linked if $A\cap B\ne\emptyset$ for all $A,B\in\mathcal L$. A linked upfamily $\mathcal M$ of subsets of $X$ is maximal linked if $\mathcal M$ coincides with each linked upfamily $\mathcal L$ on $X$ that contains $\mathcal M$. The superextension $\lambda(X)$ consists of all maximal linked upfamilies on $X$. Any associative binary operation $* : X\times X \to X$ can be extended to an associative binary operation $\circ: \lambda(X)\times\lambda(X)\to\lambda(X)$ by the formula $\mathcal L\circ\mathcal M=\Big\langle\bigcup_{a\in L}a*M_a:L\in\mathcal L,\;\{M_a\}_{a\in L}\subset\mathcal M\Big\rangle$ for maximal linked upfamilies $\mathcal{L}, \mathcal{M}\in\lambda(X)$. In the paper we describe superextensions of all three-element semigroups up to isomorphism.

On the automorphism group of the superextension of a semigroup

“…Proof. In [21,Theorem 3] it was shown that the superextension of a right (left) zero semigroup is a right (left) zero semigroup as well. Each permutation on a right (left) zero semigroup is an automorphism.…”

“…Proof. In [21,Theorem 2] it was shown that the superextension of an almost null semigroup is an almost null semigroup as well. Taking into account that z is the zero of the semigroup λ(OA X ) and a is the unique idempotent in λ(OA X ) \ {z}, we conclude that ψ(z) = z and ψ(a) = a for any ψ ∈ Aut (λ(AO X )).…”

“…Among 24 pairwise non-isomorphic semigroups of order 3 there are 12 commutative semigroups. Superextensions of all commutative semigroups of order 3 are commutative as well, see [21].…”

“…Consider the semigroups (LO 2 ) +0 , (RO 2 ) +0 , (LO 2 ) +1 and (RO 2 ) +1 . In [21] it was proved that…”

“…Introduction. In this paper we investigate the automorphism group of the superextension λ(S) of a semigroup S. The thorough study of various extensions of semigroups was started in [15] and continued in [1]- [10], [16]- [21]. The largest among these extensions is the semigroup υ(S) of all upfamilies on S. A family A of non-empty subsets of a set X is called an upfamily if for each set A ∈ A any subset B ⊃ A belongs to A.…”