2024/04/22 by Victoria Gould, Gould, Victoria, Ambroise Grau +5
Computer Science · Decision Sciences · Engineering · #20M12 #20M25 #20M30 #Advanced Control Systems Optimization #FOS: Mathematics #Fuzzy and Soft Set Theory #Rings and Algebras (math.RA) #semigroups and automata theory
paper · doi:10.48550/arxiv.2404.14005
openalex publication_date 2024/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the translational hull Ω(I) of an arbitrary subsemigroup I of an endomorphism monoid End(A) where A is a universal algebra. We give conditions for every bi-translation of I to be realised by transformations, or by endomorphisms, of A. We demonstrate that certain of these conditions are also sufficient to provide natural isomorphisms between the translational hull of I and the idealiser of I within End(A), which in the case where I is an ideal is simply End(A). We describe the connection between these conditions and work of Petrich and Gluskin in the context of densely embedded ideals. Where the conditions fail, we develop a methodology to extract information concerning Ω(I) from the translational hull Ω(I/≈) of a quotient I/≈ of I. We illustrate these concepts in detail in the cases where A is: a free algebra; an independence algebra; a finite symmetric group.