Finite Basis Of Identities Research Articles

Identities in vector spaces and examples of finite-dimensional linear algebras having no finite basis of identities

Identities in vector spaces embedded in linear algebras are considered. Issues in finite basicity of identities for vector spaces, as well as for finite-dimensional linear (nonassociative) algebras, are investigated. Essentially infinitely based and strongly infinitely based vector spaces and nonassociative linear algebras are exemplified.

An example of a simple finite-dimensional algebra with no finite basis of identities

An example of a simple finite-dimensional algebra with no finite basis of identities

ON THE VARIETY GENERATED BY THE MONOID OF TRIANGULAR 2×2 MATRICES OVER A TWO-ELEMENT FIELD

AbstractLet 𝒯n(F) denote the monoid of all upper triangularn×nmatrices over a finite fieldF. It has been shown by Volkov and Goldberg that 𝒯n(F) is nonfinitely based if ∣F∣>2 andn≥4, but the cases when ∣F∣>2 andn=2,3 or when ∣F∣=2 have remained open. In this paper, it is shown that the monoid 𝒯2(F) is finitely based when ∣F∣=2 , and a finite identity basis for it is given. Moreover, all maximal subvarieties of the variety generated by 𝒯2(F) with ∣F∣=2 are determined.

Bases of identities for semigroups of bounded rank transformations of a set

We complete the series of results by M. V. Sapir, M. V. Volkov and the author solving the Finite Basis Problem for semigroups of rank ≤ k transformations of a set, namely based on these results we prove that the semigroup Tk(X) of rank ≤ k transformations of a set X has no finite basis of identities if and only if k is a natural number and either k = 2 and |X| ∈ «3, 4» or k ≥ 3. A new method for constructing finite non-finitely based semigroups is developed. We prove that the semigroup of rank ≤ 2 transformations of a 4-element set has no finite basis of identities but that the problem of checking its identities is tractable (polynomial).

A minimal nonfinitely based semigroup whose variety is polynomially recognizable

We exhibit a 6-element semigroup that has no finite identity basis but nevertheless generates a variety whose finite membership problem admits a polynomial algorithm.

On the finite basis problem for the monoids of triangular boolean matrices

Let $${{\fancyscript{T}\fancyscript{B}_n}}$$ denote the submonoid of all upper triangular boolean n × n matrices. It was shown by Volkov and Goldberg that $${{\fancyscript{T}\fancyscript{B}_n}}$$ is nonfinitely based if n > 3, but the cases when n = 2, 3 remained open. In this paper, it is shown that the monoid $${{\fancyscript{T}\fancyscript{B}_2}}$$ is finitely based, and a finite identity basis for the monoid $${{\fancyscript{T}\fancyscript{B}_2}}$$ is given. Moreover, it is shown that $${{\fancyscript{T}\fancyscript{B}_3}}$$ is inherently nonfinitely based. Hence, $${{\fancyscript{T}\fancyscript{B}_n}}$$ is finitely based if and only if n ≤ 2.

On the finite axiomatizability of varieties of commutative unary algebras

In this paper, it is proved that every variety of commutative unary algebras of a finite signature has a finite basis of identities.

The interval [B2, NB2] in the lattice of Rees-Sushkevich varieties

We provide new solutions to the word problems for the variety B2 generated by the five element Brandt semigroup B2 with zero divisors and the variety NB2 generated by B2 and the two element left and right zero semigroups. We also provide a finite basis of identities for the variety NB2. This leads to a complete description of the interval [B2, NB2], in the lattice of semigroup varieties. This is a critical interval in the study of the lattice of aperiodic Rees-Sushkevich varieties. In addition, the varieties B2 and NB2 are especially important in the characterization of those Rees-Sushkevich varieties that are exact, that is, generated by completely 0-simple semigroups. Byproducts of this note are techniques that lend themselves to a deeper study of the structure of the free objects in the varieties in the interval [B2, NB2].

THE VARIETY GENERATED BY A CERTAIN TRANSFORMATION MONOID

It is shown that the transformation monoid [Formula: see text] is finitely based and a finite identity basis for [Formula: see text] is given, which solves the finite basis problem for [Formula: see text]. It is also shown that the variety Var[Formula: see text] generated by [Formula: see text] for which xyx is not an isoterm has uncountably many subvarieties, which gives an affirmative answer to a question of Jackson.

Flat algebras and the translation of universal Horn logic to equational logic

AbstractWe describe which subdirectly irreducible flat algebras arise in the variety generated by an arbitrary class of flat algebras with absorbing bottom element. This is used to give an elementary translation of the universal Horn logic of algebras, partial algebras, and more generally still, partial structures into the equational logic of conventional algebras. A number of examples and corollaries follow. For example, the problem of deciding which finite algebras of some fixed type have a finite basis for their quasi-identities is shown to be equivalent to the finite identity basis problem for the finite members of a finiteiy based variety with definable principal congruences.

A NEW METHOD IN THE FINITE BASIS PROBLEM WITH APPLICATIONS TO RANK 2 TRANSFORMATION SEMIGROUPS

We prove that the semigroup of all transformations of a 3-element set with rank at most 2 does not have a finite basis of identities. This gives a negative answer to a question of Shevrin and Volkov. It is worthwhile to notice that the semigroup of transformations with rank at most 2 of an n-element set, where n > 4, has a finite basis of identities. A new method of constructing finite non-finitely based semigroups is developed.

Finite orthodox locally idempotent semigroups having no finite basis of biidentities

A finitely generated existence variety of orthodox locally idempotent semigroups is constructed which has no finite basis of biidentities within the class of all orthodox semigroups. In addition, ordinary semigroup identities valid in this existence variety also cannot be finitely based. This yields examples of finite orthodox locally idempotent semigroups having both no finite basis of biidentities and also no finite basis of identities in the usual sense.

All Subvarieties of ℒpq Have Finite Bases of Identities

We consider the varieties of lattice ordered groups with the identity of commutation of the nth powers of elements. We establish that every such l-variety with n=pq, where p and q are distinct prime numbers, has a finite basis of identities.

Quasi-Identities of Congruence-Distributive Quasivarieties of Algebras

LetR be an arbitrary quasivariety of algebras and A an algebra not necessarily inR. A congruence θ ∈ ConA is an R-congruence if A/θ ∈ R. The set ConRA of all R-congruences of A makes an algebraic lattice under inclusion. A quasivariety R of algebras is congruence-distributive if the lattice of Rcongruences is distributive for every algebra A. In the case when R is a variety this definition coincides with that of a congruence-distributive variety. An algebra A in a quasivariety R is called finitely subdirectly R-indecomposable if, for every representation of A in the form of a finite subdirect product ∏ {Ai ∈ R : i < n}, one of the projections πi : A → Ai is an isomorphism. Note that if R is congruence-distributive then every finitely subdirectly R-indecomposable algebra is (absolutely) finitely subdirectly indecomposable (see [1, 2]). It is well known [3] that every congruence-distributive variety of algebras of finite signature with the finitely axiomatizable class of finitely subdirectly indecomposable algebras has a finite basis of identities. It was asked in some articles (see [1, 4, 5]) whether any congruence-distributive quasivariety R of algebras with the finitely axiomatizable class of finitely subdirectly R-indecomposable algebras has a finite basis of quasi-identities. A positive answer to this question is known in two particular cases: for finitely generated quasivarieties [1] and for quasivarieties with equationally definable intersections of principal congruences [5]. We say that a lattice L has at most countable width if every antichain in L is at most countable. A lattice L satisfies the minimality condition if every chain in L has a least element. A lattice with the maximality condition is defined by reversal. We say that a class R of algebras is finitely axiomatizable relative to a class K if there exists a finite set Ψ of propositions such that R = K ∩ Mod(Ψ). In this article we prove the following two theorems.

On the Specht Property of Varieties of Commutative Alternative Algebras over a Field of Characteristic 3 and Commutative Moufang Loops

A variety of algebras is a Specht variety if each of its subvarieties is finitely based. The problem of whether the variety of soluble alternative algebras is a Specht variety was formulated by A. M. Slin′ko in “Dnestrovskaya Tetrad′” [1, Problem 129]. This problem was solved in the affirmative in the case of a field of characteristic other than 2 and 3. As regards index 2 soluble algebras, this follows from the results of Yu. A. Medvedev’s article [2]. Furthermore, in [3] Yu. A. Medvedev exhibited a variety of soluble alternative algebras over a field of characteristic 2 without a finite basis of identities. Next, let Nk denote the variety of alternative algebras of nilpotency class at most k and A, the variety of algebras with zero multiplication. S. V. Pchelintsev [4] proved that every soluble alternative algebra A over a field of characteristic other than 2 and 3 belongs to the variety NkA ∩N3Nm; i.e., (A) = (A) = 0

Identities of locally testable semigroups

Let S be a semigroup of words over an alphabet ∑ . Suppose tliar every two words u and e over ∑ are equal in S if (1) the sets of subwords of length k of the words a and b coincide and are non-empty. (2) the prefix (suffix) of u of length k1 is equal to the prefix (suffix) of e. Then S is called k-testable. A semigroup is locally testable if it is k-testable for some k > 0. We present a finite basis of identities of the variety of A'-testable semigroups. The structure of k-testable semigroup is studied. Necessarv and sufficient conditions for local testability will be given. A solution to one problem from the survey of Shevrin and Sukhanov (1985) will be presented.

On a finite basis problem for universal positive formulas

A finite semigroup without a finite basis of collective identities has been constructed. A semigroup with a finite basis of identities, but without a finite basis of disjunctive identities has been constructed.

The join of varieties with nilpotent intersection

It is proved that the join of finitely based varieties with nilpotent intersection has a finite basis of identities.

Matrix rank 1 semigroup identities

A finite basis of identities is constructed for the semigroup of all rank 1 n × n matri­ces over the field. It is worthy to notice that every semigroup of all rank r, r > l,n×n matrices over a finite field has no finite basis of identities. Let G be an arbitrary vari­ety of groups with a finite basis of identities. A finite basis of identities is constructed for the variety generated by all completely 0-simple semigroups over G-groups.

Absence of a finite basis of identities for free solvable algebras of step 2 of finite degree of freedom

Absence of a finite basis of identities for free solvable algebras of step 2 of finite degree of freedom