The semigroup of finite partial order isomorphisms of a bounded rank of an infinite linear ordered set

TL;DR

This paper investigates the algebraic structure of the semigroup of finite partial order isomorphisms of bounded rank in an infinite linear order, revealing its stability, ideal series, and congruence properties.

Contribution

It provides a detailed description of the semigroup's algebraic properties, including idempotents, Green's relations, and the nature of its congruences, which was previously unexplored.

Findings

Semigroup is stable and has tight ideal series.

Contains only Rees' congruences and all homomorphic images have tight ideal series.

Characterizes algebraic structure of finite partial order isomorphisms in infinite linear orders.

Abstract

We study algebraic properties of the semigroup $\mathscr{O\!\!I\!}_n(L)$ of finite partial order isomorphisms of the rank $\leq n$ of an infinite linearly ordered set $(L,\leqslant)$. In particular we describe its idempotents, the natural partial order and Green's relations on $\mathscr{O\!\!I\!}_n(L)$. It is proved that the semigroup $\mathscr{O\!\!I\!}_n(L)$ is stable and it contains tight ideal series. Moreover, we show that the semigroup $\mathscr{O\!\!I\!}_n(L)$ admits only Rees' congruences and every its homomorphic image is a semigroup with tight ideal series.