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The weak extension property and finite axiomatizability for quasivarieties

100%
Dziobiak W., Maróti M., McKenzie R., Nurakunov A.
Fundamenta Mathematicae
|
2009
|
tom 202
|
nr 3
199-223
EN
We define and compare a selection of congruence properties of quasivarieties, including the relative congruence meet semi-distributivity, RSD(∧), and the weak extension property, WEP. We prove that if 𝒦 ⊆ ℒ ⊆ ℒ' are quasivarieties of finite signature, and ℒ' is finitely generated while 𝒦 ⊨ WEP, then 𝒦 is finitely axiomatizable relative to ℒ. We prove for any quasivariety 𝒦 that 𝒦 ⊨ RSD(∧) iff 𝒦 has pseudo-complemented congruence lattices and 𝒦 ⊨ WEP. Applying these results and other results proved by M. Maróti and R. McKenzie [Studia Logica 78 (2004)] we prove that a finitely generated quasivariety ℒ of finite signature is finitely axiomatizable provided that ℒ satisfies RSD(∧), or that ℒ is relatively congruence modular and is included in a residually small congruence modular variety. This yields as a corollary the full version of R. Willard's theorem for quasivarieties and partially proves a conjecture of D. Pigozzi. Finally, we provide a quasi-Maltsev type characterization for RSD(∧) quasivarieties and supply an algorithm for recognizing when the quasivariety generated by a finite set of finite algebras satisfies RSD(∧).
2
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Quasivarieties of pseudocomplemented semilattices

100%
Adams M. E., Dziobiak W., Gould M., Schmid J.
Fundamenta Mathematicae
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1994-1995
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tom 146
|
nr 3
295-312
EN
Two properties of the lattice of quasivarieties of pseudocomplemented semilattices are established, namely, in the quasivariety generated by the 3-element chain, there is a sublattice freely generated by ω elements and there are $2^ω$ quasivarieties.
3
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A deceptive fact about functions

100%
Dziobiak W., Ehrenfeucht A., Grace J. B., Silberger D.
Fundamenta Mathematicae
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2000
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tom 163
|
nr 1
83-93
EN
The paper provides a proof of a combinatorial result which pertains to the characterization of the set of equations which are solvable in the composition monoid of all partial functions on an infinite set.
4
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Finite atomistic lattices that can be represented as lattices of quasivarieties

100%
Adaricheva K. V., Dziobiak W., Gorbunov V. A.
Fundamenta Mathematicae
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1993
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tom 142
|
nr 1
19-43
EN
We prove that a finite atomistic lattice can be represented as a lattice of quasivarieties if and only if it is isomorphic to the lattice of all subsemilattices of a finite semilattice. This settles a conjecture that appeared in the context of [11].
5
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Relative congruence distributivity within quasivarieties of nearly associative Φ-algebras

98%
Dziobiak W.
Fundamenta Mathematicae
|
1990
|
tom 135
|
nr 2
77-95
6
Content available remote

Finitely generated congruence distributive quasivarieties of algebras

86%
Dziobiak W.
Fundamenta Mathematicae
|
1989
|
tom 133
|
nr 1
47-57
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