Completions for partially ordered semigroups

authored by: M. Erné, J. Z. Reichman
Abstract: A standard completion γ assigns a closure system to each partially ordered set in such a way that the point closures are precisely the (order-theoretical) principal ideals. If S is a partially ordered semigroup such that all left and all right translations are γ-continuous (i.e., Y∈γS implies {x∈S:y·x∈Y}∈γS and {x∈S:x·y∈Y}∈γS for all y∈S), then S is called a γ-semigroup. If S is a γ-semigroup, then the completion γS is a complete residuated semigroup, and the canonical principal ideal embedding of S in γS is a semigroup homomorphism. We investigate the universal properties of γ-semigroup completions and find that under rather weak conditions on γ, the category of complete residuated semigroups is a reflective subcategory of the category of γ-semigroups. Our results apply, for example, to the Dedekind-MacNeille completion by cuts, but also to certain join-completions associated with so-called "subset systems". Related facts are derived for conditional completions.
Organisation(s): Institute of Algebra, Number Theory and Discrete Mathematics
External Organisation(s): Hofstra University
Type: Article
Journal: SEMIGROUP FORUM
Volume: 34
Pages: 253-285
No. of pages: 33
ISSN: 0037-1912
Publication date: 12.1986
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Algebra and Number Theory
Electronic version(s): https://doi.org/10.1007/BF02573168 (Access: Closed)

BibTeX

@article{a6bb3ac39c344070800a6a687db7a28c,
title = "Completions for partially ordered semigroups",
abstract = "A standard completion γ assigns a closure system to each partially ordered set in such a way that the point closures are precisely the (order-theoretical) principal ideals. If S is a partially ordered semigroup such that all left and all right translations are γ-continuous (i.e., Y∈γS implies {x∈S:y·x∈Y}∈γS and {x∈S:x·y∈Y}∈γS for all y∈S), then S is called a γ-semigroup. If S is a γ-semigroup, then the completion γS is a complete residuated semigroup, and the canonical principal ideal embedding of S in γS is a semigroup homomorphism. We investigate the universal properties of γ-semigroup completions and find that under rather weak conditions on γ, the category of complete residuated semigroups is a reflective subcategory of the category of γ-semigroups. Our results apply, for example, to the Dedekind-MacNeille completion by cuts, but also to certain join-completions associated with so-called {"}subset systems{"}. Related facts are derived for conditional completions.",
author = "M. Ern{\'e} and Reichman, {J. Z.}",
year = "1986",
month = dec,
doi = "10.1007/BF02573168",
language = "English",
volume = "34",
pages = "253--285",
journal = "SEMIGROUP FORUM",
issn = "0037-1912",
publisher = "Springer New York",
number = "1",
}