Effects of semigroup properties on local embeddability
Dmitry Kudryavtsev
TL;DR
The paper investigates how local finitary properties interact with fundamental semigroup classes. It provides positive results showing that $LEF$ interacts well with completely simple and Clifford semigroups, yielding equivalences with local embeddability into finite semigroups of the same class, while proving a negative result for $\ ext{J}$-trivial semigroups by constructing a $\ ext{J}$-trivial LEF semigroup that cannot be locally embedded into finite $\ ext{J}$-trivial semigroups. To resolve differences between local embeddability and local wrappability, the authors construct non-$LEF$ semigroups that are $LWF$ using a pre-accurate/finite-quotient framework, and they present a novel family of $\ ext{J}$-trivial semigroups that satisfy $LWF$ but not $LEF$, clarifying the distinct roles of these finitary notions. The work thus refines our understanding of when LEF properties can be transferred to finite members of a given class and demonstrates a strict separation between LEF and LWF within semigroup theory. These results have implications for the model-theoretic and algorithmic aspects of semigroup approximations and provide constructive tools for further exploration of finitary approximations in non-group contexts.
Abstract
We investigate whether semigroups with a given property which are also locally embeddable into finite semigroups can be locally embedded into finite semigroups with the same property, obtaining a positive answer for completely simple and Clifford semigroups (similarly to group and inverse semigroup cases studied previously) and a negative answer for $\J$-trivial semigroups (similarly to cancellative semigroups). Additionally, we resolve the standing question on the differences between local embeddability in and local wrappability by finite structures, providing a novel construction of $\J$-trivial semigroups which satisfy the latter, but not the former.