Regular double p-algebras

2019 ◽  
Vol 69 (1) ◽  
pp. 15-34 ◽  
Author(s):  
M. E. Adams ◽  
Hanamantagouda P. Sankappanavar ◽  
Júlia Vaz de Carvalho
Keyword(s):  
Free Algebra ◽  
Explicit Description ◽  
Priestley Duality ◽  
Subdirectly Irreducible ◽  
Locally Finite ◽  
Equational Basis ◽  
Proper Subvariety ◽  
Simple Algebras ◽  
Countably Infinite ◽  
Subdirectly Irreducible Algebras

Abstract In this paper, we investigate the variety RDP of regular double p-algebras and its subvarieties RDPn, n ≥ 1, of range n. First, we present an explicit description of the subdirectly irreducible algebras (which coincide with the simple algebras) in the variety RDP1 and show that this variety is locally finite. We also show that the lattice of subvarieties of RDP1, LV(RDP1), is isomorphic to the lattice of down sets of the poset {1} ⊕ (ℕ × ℕ). We describe all the subvarieties of RDP1 and conclude that LV(RDP1) is countably infinite. An equational basis for each proper subvariety of RDP1 is given. To study the subvarieties RDPn with n ≥ 2, Priestley duality as it applies to regular double p-algebras is used. We show that each of these subvarieties is not locally finite. In fact, we prove that its 1-generated free algebra is infinite and that the lattice of its subvarieties has cardinality 2ℵ0. We also use Priestley duality to prove that RDP and each of its subvarieties RDPn are generated by their finite members.

Existentially closed algebras and boolean products

1988 ◽  
Vol 53 (2) ◽  
pp. 571-596
Author(s):  
Herbert H. J. Riedel
Keyword(s):  
Sufficient Condition ◽  
Subdirectly Irreducible ◽  
Universal Class ◽  
Model Companion ◽  
Universal Horn Class ◽  
Existentially Closed ◽  
Boolean Product ◽  
Boolean Products ◽  
Simple Algebras ◽  
Subdirectly Irreducible Algebras

AbstractA Boolean product construction is used to give examples of existentially closed algebras in the universal Horn class ISP(K) generated by a universal class K of finitely subdirectly irreducible algebras such that Γa(K) has the Fraser-Horn property. If ⟦a ≠ b⟧ ∩ ⟦c ≠ d ⟧ = ∅ is definable in K and K has a model companion of K-simple algebras, then it is shown that ISP(K) has a model companion. Conversely, a sufficient condition is given for ISP(K) to have no model companion.

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 SUBDIRECTLY IRREDUCIBLE DIFFERENTIAL MODES

2012 ◽  
Vol 22 (04) ◽  
pp. 1250028 ◽  
Author(s):  
DAVID STANOVSKÝ
Keyword(s):  
Explicit Description ◽  
Subdirectly Irreducible ◽  
Finitely Based ◽  
Congruence Lattices ◽  
Subdirectly Irreducible Algebras

We study a variety of modes (idempotent algebras with mutually commuting term operations), so called differential modes, having a strongly solvable chain 0 ≤ α ≤ 1 in their congruence lattices. We show an explicit description of subdirectly irreducible algebras in this variety, and use it to compute residual bounds of its subvarieties. It follows from our results that all subvarieties with a finite residual bound are finitely based.

Locally Finite Central Simple Algebras

2021 ◽  
Author(s):  
Tamar Bar-On ◽  
Shira Gilat ◽  
Eliyahu Matzri ◽  
Uzi Vishne
Keyword(s):  
Central Simple Algebras ◽  
Locally Finite ◽  
Simple Algebras ◽  
Central Simple

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.

SOME PROPERTIES OF FINITELY DECIDABLE VARIETIES

1992 ◽  
Vol 02 (01) ◽  
pp. 89-101 ◽  
Author(s):  
MATTHEW A. VALERIOTE ◽  
ROSS WILLARD
Keyword(s):  
Order Theory ◽  
Subdirectly Irreducible ◽  
Locally Finite ◽  
First Order ◽  
Minimal Sets ◽  
First Order Theory ◽  
Finite Member ◽  
Transfer Principles

Let [Formula: see text] be a variety whose class of finite members has a decidable first-order theory. We prove that each finite member A of [Formula: see text] satisfies the (3, 1) and (3, 2) transfer principles, and that the minimal sets of prime quotients of type 2 or 3 in A must have empty tails. The first result has already been used by J. Jeong [9] in characterizing the finite subdirectly irreducible members of [Formula: see text] with nonabelian monolith. The second result implies that if [Formula: see text] is also locally finite and omits type 1, then [Formula: see text] is congruence modular.

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.

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