scholarly journals Jankov Formula and Ternary Deductive Term

10.29007/8fkc ◽  
2018 ◽  
Author(s):  
Alex Citkin
Keyword(s):  
Finite System ◽  
Heyting Algebras ◽  
Finite Variety ◽  
Subdirectly Irreducible ◽  
Locally Finite ◽  
Defining Relations ◽  
Locally Finite Variety ◽  
Canonical Formulas ◽  
Partial Algebras

Grounding on defining relations of a finitely presentable subdirectly irreducible (s.i.) algebra in a variety with a ternary deductive term (TD), we define the characteristic identity of this algebra. For finite s.i. algebras the characteristic identity is equivalent to the identity obtained from Jankov formula. In contrast to Jankov formula, characteristic identity is relative to a variety and even in the varieties of Heyting algebras there are the characteristic identities not related to Jankov formula. Every subvariety of a given locally finite variety with a TD term admits an optimal axiomatization consisting of characteristic identities. There is an algorithm that reduces any finite system of axioms of such a variety to an optimal one. Each variety with a TD term can be axiomatized by characteristic identities of partial algebras, and in certain cases these identities are related to the canonical formulas.

FREE SPECTRA GAPS AND TAME CONGRUENCE TYPES

1995 ◽  
Vol 05 (06) ◽  
pp. 651-672 ◽  
Author(s):  
JOEL BERMAN
Keyword(s):  
Lower Bound ◽  
Finite Variety ◽  
Subdirectly Irreducible ◽  
Irreducible Algebra ◽  
Locally Finite ◽  
Finite Algebras ◽  
Locally Finite Variety ◽  
Simple Algebras ◽  
Type 3 ◽  
Free Spectra

Chapter 12 of "The Structure of Finite Algebras" by D. Hobby and R. McKenzie contains theorems revealing how the set of types appearing in a locally finite variety [Formula: see text] influences the size of the free algebra in [Formula: see text] freely generated by n elements. We provide more results in this vein. If A is a subdirectly irreducible algebra of size k, then a lower bound on the number of n-ary polynomials of A is obtained for each case that the monolith of A has type 3, 4, or 5. Examples for every k show that in each case the lower bound is the best possible. As an application of these results we show that for every finite k if all k-element simple algebras are partitioned into five classes according to their type, then algebras in each class have a sharply determined band of possible values for their free spectra. These five bands are disjoint except for some overlap on simple algebras of types 2 and 5.

scholarly journals Lattices of pseudovarieties

1989 ◽  
Vol 46 (2) ◽  
pp. 177-183 ◽  
Author(s):  
P. Agliano ◽  
J. B. Nation
Keyword(s):  
Finite Lattice ◽  
Finite Variety ◽  
Locally Finite ◽  
Finite Height ◽  
Lattice Of Subvarieties ◽  
Universal Sentence ◽  
Locally Finite Variety

AbstractWe consider the lattice of pseudovarieties contained in a given pseudovariety P. It is shown that if the lattice L of subpseudovarieties of P has finite height, then L is isomorphic to the lattice of subvarieties of a locally finite variety. Thus not every finite lattice is isomorphic to a lattice of subpseudovarieties. Moreover, the lattice of subpseudovarieties of P satisfies every positive universal sentence holding in all lattice of subvarieties of varieties V(A) ganarated by algebras A ε P.

scholarly journals The skeleton of a variety of groups

1972 ◽  
Vol 6 (3) ◽  
pp. 357-378 ◽  
Author(s):  
R.M. Bryant ◽  
L.G. Kovács
Keyword(s):  
Free Group ◽  
Finite Groups ◽  
Abelian Groups ◽  
Product Variety ◽  
Finite Variety ◽  
Locally Finite ◽  
Variety Of Groups ◽  
Locally Finite Variety ◽  
Countably Infinite ◽  
Relatively Free Group

The skeleton of a variety of groups is defined to be the intersection of the section closed classes of groups which generate . If m is an integer, m > 1, is the variety of all abelian groups of exponent dividing m, and , is any locally finite variety, it is shown that the skeleton of the product variety is the section closure of the class of finite monolithic groups in . In particular, S) generates . The elements of S are described more explicitly and as a consequence it is shown that S consists of all finite groups in if and only if m is a power of some prime p and the centre of the countably infinite relatively free group of , is a p–group.

scholarly journals Self-Rectangulating Varieties of Type 5

1997 ◽  
Vol 07 (04) ◽  
pp. 511-540 ◽  
Author(s):  
Keith A. Kearnes ◽  
Ágnes Szendrei
Keyword(s):  
Extension Property ◽  
Congruence Extension Property ◽  
Finite Variety ◽  
Finitely Generated ◽  
Locally Finite ◽  
Term Operation ◽  
Locally Finite Variety ◽  
Definable Principal Congruences

We show that a locally finite variety which omits abelian types is self-rectangulating if and only if it has a compatible semilattice term operation. Such varieties must have type-set {5}. These varieties are residually small and, when they are finitely generated, they have definable principal congruences. We show that idempotent varieties with a compatible semilattice term operation have the congruence extension property.

Varieties that make one Cross

1978 ◽  
Vol 26 (3) ◽  
pp. 368-382 ◽  
Author(s):  
Sheila Oates MacDonald ◽  
M. R. Vaughan-Lee
Keyword(s):  
Minimal Condition ◽  
Variety Of Algebras ◽  
Finite Variety ◽  
Associative Algebras ◽  
Maximal Condition ◽  
Locally Finite ◽  
Finite Algebras ◽  
Locally Finite Variety

AbstractAn example is constructed of a locally finite variety of non-associative algebras which satisfies the maximal condition on subvarieties but not the minimal condition. Based on this, counterexamples to various conjectures concerning varieties generated by finite algebras are constructed. The possibility of finding a locally finite variety of algebras which satisfies the minimal condition on subvarieties but not the maximal is also investigated.

scholarly journals Locally Finite Reducts of Heyting Algebras and Canonical Formulas

2017 ◽  
Vol 58 (1) ◽  
pp. 21-45 ◽  
Author(s):  
Guram Bezhanishvili ◽  
Nick Bezhanishvili
Keyword(s):  
Heyting Algebras ◽  
Locally Finite ◽  
Canonical Formulas

scholarly journals Canonical formulas via locally finite reducts and generalized dualities

10.29007/hgbj ◽  
2018 ◽  
Author(s):  
Nick Bezhanishvili
Keyword(s):  
Modal Logic ◽  
Intuitionistic Logic ◽  
Algebraic Approach ◽  
Modal Logics ◽  
Substructural Logic ◽  
Locally Finite ◽  
Canonical Formulas

The method of canonical formulas is a powerful tool for investigating intuitionistic and modal logics. In this talk I will discuss an algebraic approach to this method. I will mostly concentrate on the case of intuitionistic logic. But I will also review the case of modal logic and possible generalizations to substructural logic.

scholarly journals Simple surjective algebras having no proper subalgebras

1990 ◽  
Vol 48 (3) ◽  
pp. 434-454 ◽  
Author(s):  
Ágnes Szendrei
Keyword(s):  
Subdirectly Irreducible ◽  
Irreducible Algebra ◽  
Minimal Variety ◽  
Locally Finite ◽  
Subdirectly Irreducible Algebra ◽  
Matrix Power

AbstractWe prove that every finite, simple, surjective algebra having no proper subalgebras is either quasiprimal or affine or isomorphic to an algebra term equivalent to a matrix power of a unary permutational algebra. Consequently, it generates a minimal variety if and only if it is quasiprimal. We show also that a locally finite, minimal variety omitting type 1 is minimal as a quasivariety if and only if it has a unique subdirectly irreducible algebra.

scholarly journals Solution of the Word Problem for Certain Types of Groups I

1956 ◽  
Vol 3 (1) ◽  
pp. 45-54 ◽  
Author(s):  
J. L. Britton
Keyword(s):  
Finite Number ◽  
Word Problem ◽  
Infinite Number ◽  
Finite System ◽  
Defining Relations ◽  
Number Of Generators ◽  
Infinite Word ◽  
Formed Part ◽  
Free Product Of Groups ◽  
Product Of Groups

The main result of this series of papers is a theorem on the free product of groups (Theorem 1) which formed part of a doctoral thesis. This theorem has an immediate application to the word problem (Theorem 2). Usually the word problem refers to a finite system of generators and a finite number of defining relations, but in this context it is more natural to allow an infinite number of generators and defining relations. This (infinite) word problem is not solvable in general (Example 2).

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