Temat: algebraic logic - Katalog OPAC zbiorów

Skocz do pozycji: 1.

Tytuł:: Weak products of universal algebras
Autorzy:: Sain, Ildikó
Powiązania:: https://bibliotekanauki.pl/articles/1361096.pdf
Data publikacji:: 1993
Wydawca:: Polska Akademia Nauk. Instytut Matematyczny PAN
Tematy:: universal algebra
algebraic logic
cylindric algebras
Opis:: Weak direct products of arbitrary universal algebras are introduced. The usual notion for groups and rings is a special case. Some universal algebraic properties are proved and applications to cylindric and polyadic algebras are considered.
Źródło:: Banach Center Publications; 1993, 28, 1; 311-318
0137-6934
Pojawia się w:: Banach Center Publications
Dostawca treści:: Biblioteka Nauki

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Skocz do pozycji: 2.

Tytuł:: On Complete Representations and Minimal Completions in Algebraic Logic, Both Positive and Negative Results
Autorzy:: Sayed Ahmed, Tarek
Powiązania:: https://bibliotekanauki.pl/articles/2033851.pdf
Data publikacji:: 2021-07-21
Wydawca:: Uniwersytet Łódzki. Wydawnictwo Uniwersytetu Łódzkiego
Tematy:: Algebraic logic
relation algebras
cylindric algebras
polyadic algebras
complete representations
Opis:: Fix a finite ordinal \(n\geq 3\) and let \(\alpha\) be an arbitrary ordinal. Let \(\mathsf{CA}_n\) denote the class of cylindric algebras of dimension \(n\) and \(\sf RA\) denote the class of relation algebras. Let \(\mathbf{PA}_{\alpha}(\mathsf{PEA}_{\alpha})\) stand for the class of polyadic (equality) algebras of dimension \(\alpha\). We reprove that the class \(\mathsf{CRCA}_n\) of completely representable \(\mathsf{CA}_n\)s, and the class \(\sf CRRA\) of completely representable \(\mathsf{RA}\)s are not elementary, a result of Hirsch and Hodkinson. We extend this result to any variety \(\sf V\) between polyadic algebras of dimension \(n\) and diagonal free \(\mathsf{CA}_n\)s. We show that that the class of completely and strongly representable algebras in \(\sf V\) is not elementary either, reproving a result of Bulian and Hodkinson. For relation algebras, we can and will, go further. We show the class \(\sf CRRA\) is not closed under \(\equiv_{\infty,\omega}\). In contrast, we show that given \(\alpha\geq \omega\), and an atomic \(\mathfrak{A}\in \mathsf{PEA}_{\alpha}\), then for any \(n<\omega\), \(\mathfrak{Nr}_n\mathfrak{A}\) is a completely representable \(\mathsf{PEA}_n\). We show that for any \(\alpha\geq \omega\), the class of completely representable algebras in certain reducts of \(\mathsf{PA}_{\alpha}\)s, that happen to be varieties, is elementary. We show that for \(\alpha\geq \omega\), the the class of polyadic-cylindric algebras dimension \(\alpha\), introduced by Ferenczi, the completely representable algebras (slightly altering representing algebras) coincide with the atomic ones. In the last algebras cylindrifications commute only one way, in a sense weaker than full fledged commutativity of cylindrifications enjoyed by classical cylindric and polyadic algebras. Finally, we address closure under Dedekind-MacNeille completions for cylindric-like algebras of dimension \(n\) and \(\mathsf{PA}_{\alpha}\)s for \(\alpha\) an infinite ordinal, proving negative results for the first and positive ones for the second.
Źródło:: Bulletin of the Section of Logic; 2021, 50, 4; 465-511
0138-0680
2449-836X
Pojawia się w:: Bulletin of the Section of Logic
Dostawca treści:: Biblioteka Nauki

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