A LOCALLY FINITE VARIETY OF RINGS WITH AN UNDECIDABLE EQUATIONAL THEORY

2005 ◽  
Vol 57 (3) ◽  
pp. 297-307 ◽  
Author(s):  
S. Crvenkovic
Keyword(s):  
Equational Theory ◽  
Finite Variety ◽  
Locally Finite ◽  
Locally Finite Variety ◽  
Variety Of Rings

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 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.

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.

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 On the laws of the variety se

1972 ◽  
Vol 14 (3) ◽  
pp. 364-367 ◽  
Author(s):  
Roger M. Bryant
Keyword(s):  
Positive Integer ◽  
Simple Group ◽  
Finite Simple Group ◽  
Basis Property ◽  
Finite Basis ◽  
Simple Groups ◽  
Locally Finite ◽  
Prove Theorem ◽  
Finite Basis Property ◽  
Locally Finite Variety

A group is called an s-group if it is locally finite and all its Sylow subgroups are abelian. Kovács [4] has shown that, for any positive integer e, the class se of all s-groups of exponent dividing e is a (locally finite) variety. The proof of this relies on the fact that, for any e, there are only finitely many (isomorphism classes of) non-abelian finite simple groups in se; and this is a consequence of deep results of Walter and others (see [6]). In [2], Christensen raised the finite basis question for the laws of the varieties se. It is easy to establish the finite basis property for an se which contains no non-abelian finite simple group; and Christensen gave a finite basis for the laws of the variety s30, whose only non-abelian finite simple group is PSL(2,5). Here we prove Theorem For any positive integer e, the varietysehas a finite basis for its laws.

scholarly journals Condensed groups in product varieties

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Denis Osin
Keyword(s):  
Isomorphism Class ◽  
Product Variety ◽  
Equivalence Relations ◽  
Finitely Generated ◽  
Locally Finite ◽  
Elementary Equivalence ◽  
Finitely Generated Groups ◽  
Locally Finite Variety ◽  
Isolated Points ◽  
Finite Exponent

Abstract A finitely generated group 𝐺 is said to be condensed if its isomorphism class in the space of finitely generated marked groups has no isolated points. We prove that every product variety U ⁢ V \mathcal{UV} , where 𝒰 (respectively, 𝒱) is a non-abelian (respectively, a non-locally finite) variety, contains a condensed group. In particular, there exist condensed groups of finite exponent. As an application, we obtain some results on the structure of the isomorphism and elementary equivalence relations on the set of finitely generated groups in U ⁢ V \mathcal{UV} .

THE SEMIGROUPS B2 AND B0 ARE INHERENTLY NONFINITELY BASED, AS RESTRICTION SEMIGROUPS

2013 ◽  
Vol 23 (06) ◽  
pp. 1289-1335 ◽  
Author(s):  
PETER R. JONES
Keyword(s):  
Finite Basis ◽  
Brandt Semigroup ◽  
Finitely Based ◽  
Natural Structure ◽  
Locally Finite ◽  
Locally Finite Variety ◽  
Restriction Semigroup ◽  
Inverse Operation ◽  
Brandt Semigroups ◽  
Nonfinitely Based

The five-element Brandt semigroup B2 and its four-element subsemigroup B0, obtained by omitting one nonidempotent, have played key roles in the study of varieties of semigroups. Regarded in that fashion, they have long been known to be finitely based. The semigroup B2 carries the natural structure of an inverse semigroup. Regarded as such, in the signature {⋅, -1}, it is also finitely based. It is perhaps surprising, then, that in the intermediate signature of restriction semigroups — essentially, "forgetting" the inverse operation x ↦ x-1 and retaining the induced operations x ↦ x+ = xx-1 and x ↦ x* = x-1x — it is not only nonfinitely based but inherently so (every locally finite variety that contains it is also nonfinitely based). The essence of the nonfinite behavior is actually exhibited in B0, which carries the natural structure of a restriction semigroup, inherited from B2. It is again inherently nonfinitely based, regarded in that fashion. It follows that any finite restriction semigroup on which the two unary operations do not coincide is nonfinitely based. Therefore for finite restriction semigroups, the existence of a finite basis is decidable "modulo monoids". These results are consequences of — and discovered as a result of — an analysis of varieties of "strict" restriction semigroups, namely those generated by Brandt semigroups and, more generally, of varieties of "completely r-semisimple" restriction semigroups: those semigroups in which no comparable projections are related under the generalized Green relation 𝔻. For example, explicit bases of identities are found for the varieties generated by B0 and B2.

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