Cev Condition Research Articles

Braids and their monotone clones

In this paper we investigate properties of the monotone clones of certain ordered sets known asbraids. This class of ordered sets arose naturally in the study of how the clone of monotone functions on an ordered set could satisfy, or fail to satisfy, Mal'cev conditions. One version of the main result can be stated as follows. IfB is a finite braid with reachr(B)>2 (defined in the text), then the only idempotent order-preserving functionsf∶Bn→B are then projections. It then follows, for example, that no algebra of monotone functions on a finite braidB withr(B)>2 generates a congruence-modular variety.

Ideals and other generalizations of congruence classes

AbstractIn the general context of ideals in universal algebras, we study varietal properties connected with ideals that are equivalent both to Ma'cev conditions and congruence properties such as 0-regularity, 0-permutability, etc.

A Mal'cev condition for congruence principal permutable varieties

A Mal'cev condition for congruence principal permutable varieties

Horn sentences with (W) and weak Mal'cev conditions

Horn sentences with (W) and weak Mal'cev conditions

Mal'cev conditions for Horn sentences with congruence permutability

Mal'cev conditions for Horn sentences with congruence permutability

Ideals in universal algebras

In many familiar classes of algebraic structures kernels of congruence relations are uniquely specified by the inverse images q~-l(0)= {x [ q~(x)= 0} of a specified constant 0. On the one hand, q~-l(0) is nothing else but the 0-class of the kernel congruence of q~, on the other hand q~-~(0) can be axiomatized intrinsically, namely q l (0 ) is an ideal (in rings, Boolean algebras, or more generally in Heyt ing algebras), a normal subgroup, resp. normal subloop (in groups, resp. loops) or a filter (in Implicat ion algebras or Boolean algebras again, where 0 is replaced by the unit). In this paper we investigate common features of all the above structures by using a general notion of " ideal" , which makes sense in all universal algebras having a constant 0 and which specializes to the familar concepts of ideal, normal subgroup or filter in each of the algebras quoted above. In all universal algebras the 0-classes of congruence relations are easily seen to be ideals, but we shall require that conversely each ideal is the 0-class of a unique congruence relation. Such algebras, or ra ther classes of algebras with this proper ty will be called "classes with ideal determined congruences" or shortly ideal determined. In Part 1, after presenting the precise definitions, we shall show that the ideal determined varieties are characterized by a Mal 'cev condition, which turns out to be a combination of Fichtner 's condition for 0-regularity together with a ternary te rm r(x, y, z) which is a weakened form of Mal 'cev 's permutabil i ty term. From a result of Hagem ann it follows that ideal-determined varieties have modular congruence lattices, so the theory of commutators becomes readily available. In

A new product of algebras and a type reduction theorem

A construction is defined which associates, to every algebra\(\mathfrak{A}\) of a fixed but arbitrary finite similarity type, a groupoidF\(\mathfrak{A}\). The identities ofF\(\mathfrak{A}\) are finitely based if and only if those of\(\mathfrak{A}\) are, andF\(\mathfrak{A}\) is finite if and only if\(\mathfrak{A}\) is finite. Up to isomorphism,F\(\mathfrak{A}\) has the same endomorphism monoid and subalgebra lattice as\(\mathfrak{A}\), but the congruence lattice ofF\(\mathfrak{A}\) is the result of adjoining a new 1 to the congruence lattice of\(\mathfrak{A}\).F is functorial, preserves the satisfaction (and the non-satisfaction) of most Mal'cev conditions, and produces, by composition with the operation of forming the generated variety, an isomorphism of the lattice of varieties of fixed type to an interval in the lattice of varieties of groupoids.

Connections between congruence-lattices and polynomial properties

This paper gives characterisations of properties of varieties such as congruence-distributivity (CD), filtrality (FI), CD and having complemented principal congruences — these last two properties are shown to be equivalent- and having restricted equationally definable principal congruences (REDPC), in terms of the existence of some kind of polynomials. These are generalisations both of Jonsson's famous theorem characterizing CD as well as results concerning the (dual) discriminator. The methods are applied to show that REDPC implies CD, which was a problem asked in [2]. A generalisation of the concept of the Mal'cev condition — the so called Pixley-condition — is defined, and it is shown that filtrality and REDPC are Pixley-conditions. The relations between several concepts connected with the above ones are also investigated.

A Mal'cev characterization of n-permutable varieties with directly decomposable congruences

A variety V has directly decomposable congruences (abbreviated: DDCon) if every congruence relation on the product A x B of algebras A, B s V is uniquely determined by its projections onto A and B. Varieties with DDCon are characterizable by Mal'cev condition, the concrete polynomial identities were derived by G. A. Fraser and A. Horn [2] and by T.-K. Hu [unpublished]. These identities are not simple and so the phenomenon of DDCon was investigated on varieties with some additional assumptions, e.g. the Mal'cev characterization of DDCon in a permutable variety with one idempotent nullary operation is well-known, see H. Werner [9]. Recently, using H. Werner's description of permutable varieties [8], I. Chajda [1] gave a Mal'cev characterization of DDCon in permutable varieties only. We show (in Corollary 1 of this paper) that the identities of [1] can be improved; our improvement is slight in form but essential since the Mal'cev polynomial p(x = p(x, y, y) = p(y, y, x)) can be derived from the new identities and thus they characterize permutability and direct decomposability of congruences in an arbitrary variety of algebras, i.e. the assumption of permutability is redundant now (compare [1] and Corollary 1 of this paper). Making use of J. Hagemann's and A. Mitschke's description of n-permutable varieties, see [4], we generalize the above mentioned result and so a Mal'cev characterization for n-permutable varieties with DDCon is obtained. In order not to have to interrupt the discussion later, recall firstly some

Errata: Mal'cev conditions spectra and Kronecker product

I am grateful to W. Taylor for pointing out that Lemma 2.3 of Neumann (1978), which claimed that a variety and its finitary reduct have the same finite algebras, is incorrect. This weakens several results of the paper: the lattice homomorphism of Lemma 3.1 is not complete, Theorem 4.1 is false, and the Mal'cev conditions claimed to be strong in 6.1, 6.2 and 7.1 are not strong. Taylor (1980) shows one cannot improve on this.

A test for identities satisfied in lattices of submodules

SupposeR is ring with 1, andHℒ(R) denotes the variety of modular lattices generated by the class of lattices of submodules of allR-modules. An algorithm using Mal'cev conditions is given for constructing integersm≧0 andn≧1 from any given lattice polynomial inclusion formulad∈e. The main result is thatd∈e is satisfied in every lattice inHℒ(R) if and only if there existsx inR such that (m·1)x=n·1 inR, where 0·1=0 andk·1=1+1...+1 (k times) fork≧1. For example, this “divisibility” condition holds form=2 andn=1 if and only if 1+1 is an invertible element ofR, and it holds form=0 andn=12 if and only if the characteristic ofR divides 12. This result leads to a complete classification of the lattice varietiesHℒ(R),R a ring with 1. A set of representative rings is constructed, such that for each ringR there is a unique representative ringS satisfyingHℒ(R)=Hℒ(R). There is exactly one representative ring with characteristick for eachk≧1, and there are continuously many representative rings with characteristic zero. IfR has nonzero characteristic, then all free lattices inHℒ(R) have recursively solvable word problems. A necessary and sufficient condition onR is given for all free lattices inHℒ(R) to have recursively solvable word problems, ifR is a ring with characteristic zero. All lattice varieties of the formHℒ(R) are self-dual. A varietyHℒ(R) is a congruence variety, that is, it is generated by the class of congruence lattices of all members of some variety of algebras. A family of continuously many congruence varieties related to the varietiesHℒ(R) is constructed.

Mal'cev conditions, spectra and kronecker product

AbstractIt is shown that every possible spectrum of a Mal'cev definable class of varieties which should occur does occur. It follows that there are continuum many Mal'cev definable classes, a result also obtained by Taylor (1975) and Baldwin and Berman (1976).

Mal'cev conditions in sums of varieties and a new Mal'cev condition

Mal'cev conditions in sums of varieties and a new Mal'cev condition

Continuum many Mal'cev conditions

Continuum many Mal'cev conditions

A Mal’cev condition for admissible relations

A Mal’cev condition for admissible relations