Canonical Extensions and Profinite Completions of Semilattices and Lattices
Canonical extensions of (bounded) lattices have been extensively studied, and the basic existence and uniqueness theorems for these have been extended to general posets. This paper focuses on the intermediate class {Mathematical expression} of (unital) meet semilattices. Any {Mathematical expression...
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| Format: | Journal article |
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2014
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| _version_ | 1797083677703798784 |
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| author | Gouveia, M Priestley, H |
| author_facet | Gouveia, M Priestley, H |
| author_sort | Gouveia, M |
| collection | OXFORD |
| description | Canonical extensions of (bounded) lattices have been extensively studied, and the basic existence and uniqueness theorems for these have been extended to general posets. This paper focuses on the intermediate class {Mathematical expression} of (unital) meet semilattices. Any {Mathematical expression} embeds into the algebraic closure system Filt(Filt(S)). This iterated filter completion, denoted Filt(S), is a compact and {Mathematical expression}-dense extension of S. The complete meet-subsemilattice S of Filt(S) consisting of those elements which satisfy the condition of {Mathematical expression}-density is shown to provide a realisation of the canonical extension of S. The easy validation of the construction is independent of the theory of Galois connections. Canonical extensions of bounded lattices are brought within this framework by considering semilattice reducts. Any S in {Mathematical expression} has a profinite completion, {Mathematical expression}. Via the duality theory available for semilattices, {Mathematical expression} can be identified with Filt(S), or, if an abstract approach is adopted, with {Mathematical expression}, the free join completion of the free meet completion of S. Lifting of semilattice morphisms can be considered in any of these settings. This leads, inter alia, to a very transparent proof that a homomorphism between bounded lattices lifts to a complete lattice homomorphism between the canonical extensions. Finally, we demonstrate, with examples, that the profinite completion of S, for {Mathematical expression}, need not be a canonical extension. This contrasts with the situation for the variety of bounded distributive lattices, within which profinite completion and canonical extension coincide. © 2013 Springer Science+Business Media Dordrecht. |
| first_indexed | 2024-03-07T01:44:50Z |
| format | Journal article |
| id | oxford-uuid:980d1d23-36c7-4d6e-aba2-5a36e8286a9f |
| institution | University of Oxford |
| last_indexed | 2024-03-07T01:44:50Z |
| publishDate | 2014 |
| record_format | dspace |
| spelling | oxford-uuid:980d1d23-36c7-4d6e-aba2-5a36e8286a9f2022-03-27T00:04:21ZCanonical Extensions and Profinite Completions of Semilattices and LatticesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:980d1d23-36c7-4d6e-aba2-5a36e8286a9fSymplectic Elements at Oxford2014Gouveia, MPriestley, HCanonical extensions of (bounded) lattices have been extensively studied, and the basic existence and uniqueness theorems for these have been extended to general posets. This paper focuses on the intermediate class {Mathematical expression} of (unital) meet semilattices. Any {Mathematical expression} embeds into the algebraic closure system Filt(Filt(S)). This iterated filter completion, denoted Filt(S), is a compact and {Mathematical expression}-dense extension of S. The complete meet-subsemilattice S of Filt(S) consisting of those elements which satisfy the condition of {Mathematical expression}-density is shown to provide a realisation of the canonical extension of S. The easy validation of the construction is independent of the theory of Galois connections. Canonical extensions of bounded lattices are brought within this framework by considering semilattice reducts. Any S in {Mathematical expression} has a profinite completion, {Mathematical expression}. Via the duality theory available for semilattices, {Mathematical expression} can be identified with Filt(S), or, if an abstract approach is adopted, with {Mathematical expression}, the free join completion of the free meet completion of S. Lifting of semilattice morphisms can be considered in any of these settings. This leads, inter alia, to a very transparent proof that a homomorphism between bounded lattices lifts to a complete lattice homomorphism between the canonical extensions. Finally, we demonstrate, with examples, that the profinite completion of S, for {Mathematical expression}, need not be a canonical extension. This contrasts with the situation for the variety of bounded distributive lattices, within which profinite completion and canonical extension coincide. © 2013 Springer Science+Business Media Dordrecht. |
| spellingShingle | Gouveia, M Priestley, H Canonical Extensions and Profinite Completions of Semilattices and Lattices |
| title | Canonical Extensions and Profinite Completions of Semilattices and Lattices |
| title_full | Canonical Extensions and Profinite Completions of Semilattices and Lattices |
| title_fullStr | Canonical Extensions and Profinite Completions of Semilattices and Lattices |
| title_full_unstemmed | Canonical Extensions and Profinite Completions of Semilattices and Lattices |
| title_short | Canonical Extensions and Profinite Completions of Semilattices and Lattices |
| title_sort | canonical extensions and profinite completions of semilattices and lattices |
| work_keys_str_mv | AT gouveiam canonicalextensionsandprofinitecompletionsofsemilatticesandlattices AT priestleyh canonicalextensionsandprofinitecompletionsofsemilatticesandlattices |