Details
Original language | English |
---|---|
Pages (from-to) | 253-285 |
Number of pages | 33 |
Journal | SEMIGROUP FORUM |
Volume | 34 |
Issue number | 1 |
Publication status | Published - 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
- Mathematics(all)
- Algebra and Number Theory
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In: SEMIGROUP FORUM, Vol. 34, No. 1, 12.1986, p. 253-285.
Research output: Contribution to journal › Article › Research › peer review
}
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 -