Trivial Variety Research Articles

Kernel classes of varieties of completely regular semigroups I

Several complete congruences on the lattice $${\mathcal {L}}({{\mathcal {C}}}{{\mathcal {R}}})$$ of varieties of completely regular semigroups have been fundamental to studies of the structure of $${\mathcal {L}}({{\mathcal {C}}}{{\mathcal {R}}})$$. These are the kernel relation K, the left trace relation $$T_{\ell }$$, the right trace relation $$T_r$$ and their intersections $$K\cap T_{\ell }, K\cap T_r$$. However, with the exception of the lattice of all band varieties which happens to coincide with the kernel class of the trivial variety, almost nothing is known about the internal structure of individual K classes beyond the fact that they are intervals in $${\mathcal {L}}({{\mathcal {C}}}{{\mathcal {R}}})$$ (with the notable exception of the remarkable results in the very recent article by Kad’ourek (Int J Algebra Comput, https://doi.org/10.1142/S0218196719500541, 2019)). Here we present a number of general results that are pertinent to the study of K classes. This includes a variation of the renowned Polak Theorem and its relationship to the complete retraction $${\mathcal {V}} \longrightarrow {\mathcal {V}}\cap {\mathcal {B}}$$, where $${\mathcal {B}}$$ denotes the variety of bands. These results are then applied, here and in a sequel, to the detailed analysis of certain families of K classes. The paper concludes with results hinting at the complexity of K classes in general, such as that the classes of relation $$K/(K\cap T_{\ell })$$ may have the cardinality of the continuum.

Kernel classes of varieties of completely regular semigroups II

Several complete congruences on the lattice $${\mathcal {L}}({{\mathcal {C}}}{{\mathcal {R}}})$$ of varieties of completely regular semigroups have been fundamental to studies of the structure of $${\mathcal {L}}({{\mathcal {C}}}{{\mathcal {R}}})$$. These are the kernel relation K, the left trace relation $$T_{\ell }$$, the right trace relation $$T_r$$ and their intersections $$K\cap T_{\ell }, K\cap T_r$$. However, with the exception of the lattice of all band varieties which happens to coincide with the kernel class of the trivial variety, almost nothing is known about the internal structure of individual K classes beyond the fact that they are intervals in $${\mathcal {L}}({{\mathcal {C}}}{{\mathcal {R}}})$$ (with the notable exception of the remarkable results in the very recent article by Kad’ourek (Int J Algebra Comput, https://doi.org/10.1142/S0218196719500541, 2019)). Here we present a number of general results that are pertinent to the study of K classes. This includes a variation of the renowned Polak Theorem and its relationship to the complete retraction $${\mathcal {V}} \longrightarrow {\mathcal {V}}\cap {\mathcal {B}}$$, where $${\mathcal {B}}$$ denotes the variety of bands. These results are then applied, here and in a sequel, to the detailed analysis of certain families of K classes. The paper concludes with results hinting at the complexity of K classes in general, such as that the classes of relation $$K/(K\cap T_{\ell })$$ may have the cardinality of the continuum.

Universal Triviality of the Chow Group of 0-cycles and the Brauer Group

AbstractWe prove that a smooth proper universally $\textrm{CH}_0$-trivial variety $X$ over a field $k$ has universally trivial Brauer group. This fills a gap in the literature concerning the $p$-torsion of the Brauer group when $k$ has characteristic $p$.

Varieties generated by unital Abelian ℓ-groups

Much work has been done on the connections between Abelian ℓ-groups and MV-algebras. For example, it is well known that there is a bijective correspondence between the so-called equational classes of unital Abelian ℓ-groups and the subvarieties of MV, the variety of MV-algebras. The drawback of this correspondence is that these former classes are not varieties. The main result of this paper remedies this situation by establishing that the subvariety lattice of the variety pA of positively-pointed Abelian ℓ-groups (excluding the trivial variety) is isomorphic to the subvariety lattice of MV. In particular, with the exception of the unique atom, every subvariety of pA is generated by its unital members.

Support Varieties for Modules Over Stacked Monomial Algebras

Let Λ be a finite-dimensional (D, A)-stacked monomial algebra. In this article, we give necessary and sufficient conditions for the variety of a simple Λ-module to be nontrivial. This is then used to give structural information on the algebra Λ, as it is shown that if the variety of every simple module is nontrivial, then Λ is a D-Koszul monomial algebra. We also provide examples of (D, A)-stacked monomial algebras which are not self-injective but nevertheless satisfy the finite generation conditions (Fg1) and (Fg2) of [4], from which we can characterize all modules with trivial variety.

Another independent self-dual basis for the trivial variety

We give a new independent self-dual 3-basis for the trivial variety with two binary operations.

Cancellative residuated lattices

Cancellative residuated lattices are natural generalizations of lattice-ordered groups ( $$ \mathcal{l} $$ -groups). Although cancellative monoids are defined by quasi-equations, the class $$ \mathcal{CanRL} $$ of cancellative residuated lattices is a variety. We prove that there are only two commutative subvarieties of $$ \mathcal{CanRL} $$ that cover the trivial variety, namely the varieties generated by the integers and the negative integers (with zero). We also construct examples showing that in contrast to $$ \mathcal{l} $$ -groups, the lattice reducts of cancellative residuated lattices need not be distributive. In fact we prove that every lattice can be embedded in the lattice reduct of a cancellative residuated lattice. Moreover, we show that there exists an order-preserving injection of the lattice of all lattice varieties into the subvariety lattice of $$ \mathcal{CanRL} $$ . We define generalized MV-algebras and generalized BL-algebras and prove that the cancellative integral members of these varieties are precisely the negative cones of $$ \mathcal{l} $$ -groups, hence the latter form a variety, denoted by $$ \mathcal{LG}^- $$ . Furthermore we prove that the map that sends a subvariety of $$ \mathcal{l} $$ -groups to the corresponding class of negative cones is a lattice isomorphism from the lattice of subvarieties of $$ \mathcal{LG}$$ to the lattice of subvarieties of $$ \mathcal{LG}^- $$ . Finally, we show how to translate equational bases between corresponding subvarieties, and briefly discuss these results in the context of R. McKenzie’s characterization of categorically equivalent varieties.

Atomic varieties of sets with relative inverses

The generalization of the construction of the lattice of varieties for partial algebras is used for sets with relative inverses. There are many quantum structures representable by sets with relative inverses (orthomodular lattices, orthoalgebras, D-posets, test spaces,...). Varieties covering the trivial variety are investigated for the case of closed (strongest type) subalgebras and closed homomorphisms. Some similar results for weaker types are given. The context with set representation problems is considered for the set-theoretic difference operations.

The modularity of the lattice of varieties of completely regular semigroups and related representations

A semigroup endowed with a unary operation satisfying the identitiesis a completely regular semigroup. In several recent papers devoted to the study of the lattice of subvarieties of the variety of completely regular semigroups, various results have been obtained which decompose special intervals in into either direct products or subdirect products. Petrich [14], Hall and Jones [6] and Rasin [20] have shown that certain intervals of the form , where is the trivial variety and are subdirect products of and Pastijn and Trotter [13] show that certain intervals of the form are direct products of the intervals and The main objective of this paper is to develop an appropriate lattice theoretic framework for these representations.

Decidability problem for finite Heyting algebras

AbstractThe aim of this paper is to characterize varieties of Heyting algebras with decidable theory of their finite members. Actually we prove that such varieties are exactly the varieties generated by linearly ordered algebras. It contrasts to the result of Burris [2] saying that in the case of whole varieties, only trivial variety and the variety of Boolean algebras have decidable first order theories.

Semigroups generated by certain operators on varieties of completely regular semigroups

The operators C,K,L,T,T( and Tr on the lattice Se(^0t) of varieties of completely regular semigroups have played an important role in recent studies of S^(^9t Although each of these operators is idempotent, when applied in various combinations to the trivial variety they yield varieties for which the only upper bound is ^^. The semigroups generated by various subsets of (C, K, L, T, Tr, T{) are determined here in terms of generators and relations. 1. Introduction and summary. Completely regular semigroups (unions of groups) may be regarded as algebras with the operations of (binary) multiplication and (unary) inversion. As such they form a variety <&{% defined by the identities (1) (ab)c = (ab)c, a = aa~ιa, aa'1 = a~ιa, (a~ι)~ = a. The lattice ££{^01) of all subvarieties of <£!% turns out to be amenable to a thorough analysis both globally and locally. The former includes various (complete) congruences that emerge naturally in the study either of the varieties themselves or of the corresponding fully invariant congruences on a free completely regular semigroup FΉffl on a countably infinite set. Local studies of the lattice SP^Si) usually amount to rather complete descriptions of relatively small intervals in JS?(^^) modulo «S?(^), the lattice of group varieties, starting from the bottom of the lattice. In the local approach, a number of operators make their appearance in the description of certain varieties in terms of some of their proper subvarieties. But these operators may be defined on all of «£?(^^) thereby providing a certain amount of information for varieties scattered throughout S^{^0i) and hence may be used for a global study of this lattice. Another source of operators on ^(Ήέft) are the kernel and trace relations on the lattice of fully invariant congruences on FΉffl now translated into relations on JS?(^^). Of the considerable literature on varieties of completely regular semigroups, we mention only the following ones because they are directly related to our object of study. We thus cite Jones [6], [7], Kadourek [8],

Homomorphisms of distributive p-algebras with countably many minimal prime ideals

According to a result of Lee, varieties of pseudocomplemented distributive lattices form an ω+1 chain in which is the trivial variety and is the variety of Boolean algebras. In the present paper it is shown that the variety contains an almost universal subcategory B in which the members of Hom(B,B') associated with minimal prime ideals of B form a countably infinite set for any B,B' ∈ B. In particular, B3contains arbitrarily large algebras whose nontrivial endomorphisms form the countably infinite right zero semigroup. Our earlier results concerning categorical properties of varieties of pseudocomplemented distributive lattices show that no further reduction of the right zero count is possible.

The subvariety lattice of the variety of distributive doublep-algebras

LetLdenote the subvariety lattice of the variety of distributive doublep-algebras, that is, the lattice whose universe consists of all varieties of distributive doublep-algebras and whose ordering is the inclusion relation. We prove in this paper that each proper filter inLis uncountable. Moreover, we prove that except for the trivial variety (the zero inL) and the variety of Boolean algebras (the unique atom inL) every other element ofL, generated by a finite algebra, has infinitely many covers inL, among which at least one is not generated by any finite algebra. The former result strengthens a result of Urquhart who showed that the latticeLis uncountable. On the other hand, both of our results indicate a high complexity of the latticeLat least in comparison with the subvariety lattice of the variety of distributivep-algebras, since a result of Lee shows that the latter lattice forms a chain of type ω + 1 and every cover in it of the variety generated by a finite algebra is itself generated by a finite algebra.