Embedding properties of endomorphism semigroups

TL;DR

This paper investigates the embedding properties of endomorphism semigroups and related structures, establishing cardinality bounds and non-embeddability results that answer open questions in algebra and semigroup theory.

Contribution

It provides new results on when certain endomorphism semigroups can or cannot embed into their duals, with optimal cardinality bounds, addressing open problems in algebra.

Findings

Self(X) embeds into dual of Self(Y) iff card(Y) >= 2^card(X)

No embedding from End(V) into its dual for infinite-dimensional V

End(F) has no embedding into its dual if F has fewer than 2^card(X) operations

Abstract

Denote by PSelf(X) (resp., Self(X)) the partial (resp., full) transformation monoid over a set X, and by Sub(V) (resp., End(V)) the collection of all subspaces (resp., endomorphisms) of a vector space V. We prove various results that imply the following: (1) If X has at least two elements, then Self(X) has a semigroup embedding into the dual of Self(Y) iff card(Y) >= 2^card(X). In particular, if X has at least two elements, then there exists no semigroup embedding from Self(X) into the dual of PSelf(X). (2) If V is infinite-dimensional, then there are no embedding from (Sub(V),+) into (Sub(V),\cap) and no semigroup embedding from End(V) into its dual. (3) Let F be an algebra freely generated by an infinite subset X. If F has less than 2^card(X) operations, then End(F) has no semigroup embedding into its dual. The cardinality bound 2^card(X) is optimal. (4) Let F be a free left module over a left aleph one - noetherian ring (i.e., a ring without strictly increasing chains, of length aleph one, of left ideals). Then End(F) has no semigroup embedding into its dual. (1) and (2) above solve questions proposed by B. M. Schein and G. M. Bergman. We also formalize our results in the settings of algebras endowed with a notion of independence (in particular independence algebras).