Loading [MathJax]/extensions/tex2jax.js

Completions for partially ordered semigroups

Research output: Contribution to journalArticleResearchpeer review

Authors

  • M. Erné
  • J. Z. Reichman

External Research Organisations

  • Hofstra University

Details

Original languageEnglish
Pages (from-to)253-285
Number of pages33
JournalSEMIGROUP FORUM
Volume34
Issue number1
Publication statusPublished - Dec 1986

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.

ASJC Scopus subject areas

Cite this

Completions for partially ordered semigroups. / Erné, M.; Reichman, J. Z.
In: SEMIGROUP FORUM, Vol. 34, No. 1, 12.1986, p. 253-285.

Research output: Contribution to journalArticleResearchpeer review

Erné, M & Reichman, JZ 1986, 'Completions for partially ordered semigroups', SEMIGROUP FORUM, vol. 34, no. 1, pp. 253-285. https://doi.org/10.1007/BF02573168
Erné M, Reichman JZ. Completions for partially ordered semigroups. SEMIGROUP FORUM. 1986 Dec;34(1):253-285. doi: 10.1007/BF02573168
Erné, M. ; Reichman, J. Z. / Completions for partially ordered semigroups. In: SEMIGROUP FORUM. 1986 ; Vol. 34, No. 1. pp. 253-285.
Download
@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",

}

Download

TY - JOUR

T1 - Completions for partially ordered semigroups

AU - Erné, M.

AU - Reichman, J. Z.

PY - 1986/12

Y1 - 1986/12

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=51649152395&partnerID=8YFLogxK

U2 - 10.1007/BF02573168

DO - 10.1007/BF02573168

M3 - Article

AN - SCOPUS:51649152395

VL - 34

SP - 253

EP - 285

JO - SEMIGROUP FORUM

JF - SEMIGROUP FORUM

SN - 0037-1912

IS - 1

ER -