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Varieties with polynomially many models, I

100%
Idziak P., McKenzie R.
Fundamenta Mathematicae
|
2001
|
tom 170
|
nr 1-2
53-68
EN
A characterization of locally finite congruence modular varieties with the number of at most k-generated models being bounded from above by a polynomial in k is given. These are exactly the varieties polynomially equivalent to the varieties of unitary modules over a finite ring of finite representation type.
2
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The weak extension property and finite axiomatizability for quasivarieties

81%
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(∧).
3
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Cardinal multiplication of structures with a reflexive relation

70%
McKenzie R.
Fundamenta Mathematicae
|
1971
|
tom 70
|
nr 1
59-101
4
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$ℵ_1$-incompactness of Z

50%
McKenzie R.
Colloquium Mathematicum
|
1971
|
tom 23
|
nr 2
199-202
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