Characterizing semigroups $X$ with commutative extensions $\varphi(X)$, $\lambda(X)$, $N_2(X)$, $\upsilon(X)$

TL;DR

This paper characterizes the conditions under which various semigroup extensions, including filters, linked systems, and upfamilies, are commutative, providing a comprehensive understanding of their algebraic structure.

Contribution

It offers a complete characterization of semigroups with commutative extensions across multiple related constructs, advancing the algebraic theory of semigroup extensions.

Findings

Semigroups with commutative filter semigroups are characterized.

Conditions for commutativity in maximal linked systems are identified.

Results unify various semigroup extension properties under a common framework.

Abstract

We characterize semigroups $X$ whose semigroups of filters $\varphi(X)$, maximal linked systems $\lambda(X)$, linked upfamilies $N_2(X)$, and upfamilies $\upsilon(X)$ are commutative.