Search

In that paper we study atoms of congruence lattices and subdirectly irreducibility of algebras with one operator and the main symmetric operation. A ternary operation 𝑑(𝑥, 𝑦, 𝑧) satisfying identities 𝑑(𝑥, 𝑦, 𝑦) = 𝑑(𝑦, 𝑦, 𝑥) = 𝑑(𝑦, 𝑥, 𝑦) = 𝑥 is called a minority operation. The symmetric operation is a minority operation defined by specific way. An algebra 𝐴 is called subdirectly irreducible if 𝐴 has the smallest nonzero congruence. An algebra with operators is an universal algebra whose signature consists of two nonempty non-intersectional parts: the main one which can contain arbitrary operations, and the additional one consisting of operators. The operators are unary operations that act as endomorphisms with respect to the main operations, i.e., one that permutable with main operations. A lattice 𝐿 with zero is called atomic if any element of 𝐿 contains some atom. A lattice 𝐿 with zero is called atomistic if any nonzero element of 𝐿 is a join of some atom set. It shown that congruence lattices of algebras with one operator and main symmetric operation are atomic. The structure of atoms in the congruence lattices of algebras in given class is described. The full describe of subdirectly irreducible algebras and of algebras with an atomistic congruence lattice in given class is obtained.

We prove that if an n-element algebra generates the variety $$\mathcal {V}$$ which is actively structurally complete, then the cardinality of the carrier of each subdirectly irreducible algebra in $$\mathcal {V}$$ is at most $$n^{(n+1)\cdot n^{2\cdot n}}$$. As a consequence, with the use of known results, we show that there exist algorithms deciding whether a given finite algebra $$\mathbf {A}$$ generates the (actively) structurally complete variety $${\textsf {V}}(\mathbf {A})$$ in the cases when $${\textsf {V}}(\mathbf {A})$$ is congruence modular or $${\textsf {V}}(\mathbf {A})$$ is congruence meet-semidistributive or $$\mathbf {A}$$ is a semigroup.

In this paper, we investigate the variety RDP of regular double p-algebras and its subvarieties RDP n , 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 RDP 1 and show that this variety is locally finite. We also show that the lattice of subvarieties of RDP 1, LV (RDP 1), is isomorphic to the lattice of down sets of the poset {1} ⊕ (ℕ × ℕ). We describe all the subvarieties of RDP 1 and conclude that LV (RDP 1) is countably infinite. An equational basis for each proper subvariety of RDP 1 is given. To study the subvarieties RDP n 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 2 ℵ 0 . We also use Priestley duality to prove that RDP and each of its subvarieties RDP n are generated by their finite members.

A commutative residuated lattice $${\varvec{A}}$$ is said to be subidempotent if the lower bounds of its neutral element e are idempotent (in which case they naturally constitute a Brouwerian algebra $${\varvec{A}}^-$$ ). It is proved here that epimorphisms are surjective in a variety $${\mathsf {K}}$$ of such algebras $${\varvec{A}}$$ (with or without involution), provided that each finitely subdirectly irreducible algebra $${\varvec{B}}\in {\mathsf {K}}$$ has two properties: (1) $${\varvec{B}}$$ is generated by lower bounds of e, and (2) the poset of prime filters of $${\varvec{B}}^-$$ has finite depth. Neither (1) nor (2) may be dropped. The proof adapts to the presence of bounds. The result generalizes some recent findings of G. Bezhanishvili and the first two authors concerning epimorphisms in varieties of Brouwerian algebras, Heyting algebras and Sugihara monoids, but its scope also encompasses a range of interesting varieties of De Morgan monoids.

In this paper, we investigate the varieties $\mathbf M_n$ and $\mathbf K_n$ of regular pseudocomplemented de Morgan and Kleene algebras of range $n$, respectively. Priestley duality as it applies to pseudocomplemented de Morgan algebras is used. We characterise the dual spaces of the simple (equivalently, subdirectly irreducible) algebras in $\mathbf M_n$ and explicitly describe the dual spaces of the simple algebras in $\mathbf M_1$ and $\mathbf K_1$. We show that the variety $\mathbf M_1$ is locally finite, but this property does not extend to $\mathbf M_n$ or even $\mathbf K_n$ for $n \geq 2$. We also show that the lattice of subvarieties of $\mathbf K_1$ is an $\omega + 1$ chain and the cardinality of the lattice of subvarieties of either $\mathbf K_2$ or $\mathbf M_1$ is $2^{\omega}$. A description of the lattice of subvarieties of $\mathbf M_1$ is given., Comment: 29 pages; 2 figures

The monoid Cb of name substitutions originated by Pitts in name abstraction, and the notion of a finitely supported Cb-set appeared as an important model for semantics of programming languages in the works of Gabbay and Pitts.The aim of this paper is to study and characterize simple, subdirectly irreducible, and indecomposable finitely supported Cb-sets. To study and find their relations, we introduce the notion of fix-simple Cb-sets, and completely describe them. Among other things, we show that under some conditions on supports, fix-simple finitely supported Cb-sets are exactly the ones whose congruence lattices are the two or three element chains. Also, it is shown that an infinite finitely supported Cb-set is simple if and only if it is a special fix-simple one. Furthermore, we observe that all the elements of a simple or subdirectly irreducible infinite Cb-set are finitely supported if we have enough elements with finite support.

In this short note we show that given a variety V of FL e w -algebras generated by a well-connected algebra, there always exists a subdirectly irreducible algebra generating V . This solves an open problem first raised in Galatos et al., Residuated lattices: an algebraic glimpse at substructural logics (2007).

An algebraic structure A is said to be finitely subdirectly reducible if A is not finitely subdirectly irreducible. We show that for any signature providing only finitely many relation symbols, the class of finitely subdirectly reducible algebraic structures is closed with respect to the formation of ultraproducts. We provide some corollaries and examples for axiomatizable classes that are closed with respect to the formation of finite subdirect products, in particular, for varieties and quasivarieties.

In this paper, we use Block’s results to classify irreducible modules over the diamond Lie algebra 𝔇. As a corollary, we also give a classification of irreducible modules over the Euclidean algebra...

The IKt-algebras that we investigate in this paper were introduced in the paper An algebraic axiomatization of the Ewald’s intuitionistic tense logic by the first and third author. Now we characterize by topological methods the subdirectly irreducible IKt-algebras and particularly the simple IKt-algebras. Finally, we consider the particular cases of finite IKt-algebras and complete IKt-algebras.

For [Formula: see text] a positive integer and [Formula: see text] a finite set of finite algebras, let [Formula: see text] denote the largest [Formula: see text]-generated subdirect product whose subdirect factors are algebras in [Formula: see text]. When [Formula: see text] is the set of all [Formula: see text]-generated subdirectly irreducible algebras in a locally finite variety [Formula: see text], then [Formula: see text] is the free algebra [Formula: see text] on [Formula: see text] free generators for [Formula: see text]. For a finite algebra [Formula: see text] the algebra [Formula: see text] is the largest [Formula: see text]-generated subdirect power of [Formula: see text]. For every [Formula: see text] and finite [Formula: see text] we provide an upper bound on the cardinality of [Formula: see text]. This upper bound depends only on [Formula: see text] and these basic parameters: the cardinality of the automorphism group of [Formula: see text], the cardinalities of the subalgebras of [Formula: see text], and the cardinalities of the equivalence classes of certain equivalence relations arising from congruence relations of [Formula: see text]. Using this upper bound on [Formula: see text]-generated subdirect powers of [Formula: see text], as [Formula: see text] ranges over the [Formula: see text]-generated subdirectly irreducible algebras in [Formula: see text], we obtain an upper bound on [Formula: see text]. And if all the [Formula: see text]-generated subdirectly irreducible algebras in [Formula: see text] have congruence lattices that are chains, then we characterize in several ways those [Formula: see text] for which this upper bound is obtained.

The main goal of this paper is to investigate very true MTL-algebras and prove the completeness of the very true MTL-logic. In this paper, the concept of very true operators on MTL-algebras is introduced and some related properties are investigated. Also, conditions for an MTL-algebra to be an MV-algebra and a Gödel algebra are given via this operator. Moreover, very true filters on very true MTL-algebras are studied. In particular, subdirectly irreducible very true MTL-algebras are characterized and an analogous of representation theorem for very true MTL-algebras is proved. Then, the left and right stabilizers of very true MTL-algebras are introduced and some related properties are given. As applications of stabilizer of very true MTL-algebras, we produce a basis for a topology on very true MTL-algebras and show that the generated topology by this basis is Baire, connected, locally connected and separable. Finally, the corresponding logic very true MTL-logic is constructed and the soundness and completeness of this logic are proved based on very true MTL-algebras.

The variety DMM of De Morgan monoids has just four minimal subvarieties. The join-irreducible covers of these atoms in the subvariety lattice of DMM are investigated. One of the two atoms consisting of idempotent algebras has no such cover; the other has just one. The remaining two atoms lack nontrivial idempotent members. They are generated, respectively, by 4-element De Morgan monoids C4 and D4, where C4 is the only nontrivial 0-generated algebra onto which finitely subdirectly irreducible De Morgan monoids may be mapped by noninjective homomorphisms. The homomorphic preimages of C4 within DMM (together with the trivial De Morgan monoids) constitute a proper quasivariety, which is shown to have a largest subvariety U. The covers of the variety (C4) within U are revealed here. There are just ten of them (all finitely generated). In exactly six of these ten varieties, all nontrivial members have C4 as a retract. In the varietal join of those six classes, every subquasivariety is a variety—in fact, every finite subdirectly irreducible algebra is projective. Beyond U, all covers of (C4) [or of (D4)] within DMM are discriminator varieties. Of these, we identify infinitely many that are finitely generated, and some that are not. We also prove that there are just 68 minimal quasivarieties of De Morgan monoids.

Basic algebras form a common generalization of MV-algebras and of orthomodular lattices, the algebraic tool for axiomatization of many-valued Łukasiewicz logic and the logic of quantum mechanics. Hence, they are included among the socalled quantum structures. An important role is played by linearly ordered basic algebras because every subdirectly irreducible MV-algebra and every subdirectly irreducible commutative basic algebra is linearly ordered. Since subdirectly irreducible linearly ordered basic algebras exist of any infinite cardinality, the natural question is to describe all possible order types of these algebras. This problem is solved in the paper.

An n-ary local polymorphism of a given monounary algebra A is a homomorphism from a finitely generated subalgebra of An to A. A is n-polymorphism-homogeneous if each n-ary local polymorphism can be extended to a global polymorphism. Then A is called polymorphism-homogeneous, if it is n-polymorphism-homogeneous for each positive integer n. In this paper we describe all polymorphism-homogeneous monounary algebras.

A theorem of single-sorted universal algebra asserts that every finite algebra can be represented as a product of a finite family of finite directly irreducible algebras. In this article, we show that the many-sorted counterpart of the above theorem is also true, but under the condition of requiring, in the definition of directly reducible many-sorted algebra, that the supports of the factors should be included in the support of the many-sorted algebra. Moreover, we show that the theorem of Birkhoff, according to which every single-sorted algebra is isomorphic to a subdirect product of subdirectly irreducible algebras, is also true in the field of many-sorted algebras.

The goal of this paper is to generalize a notion of quasi-characteristic inference rule (by using finite partial algebras instead of finite subdirectly irreducible algebras) in the following way: with every finite partial algebra we associate a (multiple-conclusion) rule, and study the properties of these rules. We prove that any equivalential logic can be axiomatized by such rules. We further discuss the correlations between characteristic rules of the finite partial algebras and canonical rules. Then, with every algebra we associate a set of characteristic rules that correspond to each finite partial subalgebra of this algebra. Finally, we demonstrate that in many respects these sets enjoy the same properties as regular quasi-characteristic rules.

In this note, we describe the irreducible representations and give a dimension formula for the Framisation of the Temperley–Lieb algebra.

Subdirect decomposition of algebra is one of its quite general and important constructions. In this paper, some subdirect decompositions (including subdirect irreducible decompositions) of finite distributive lattices and finite chains are studied, and some general results are obtained.

The Zariski closure of an arbitrary representable (not necessarily associative) algebra is studied in the general context of universal algebra, with an application being that the codimension sequence is exponentially bounded.

In this work we review the class T of ternary algebras introduced by J. A. Brzozowski and C. J. Serger in [1]. We determine properties of the congruence lattice of a ternary algebra A. The most important result refers to the construction of the free ternary algebra on a poset. In particular, we describe the poset of the join irreducible elements of the free ternary algebra with two free generators.

David Hilbert’s solvability criterion for polynomial systems in n variables from the 1890s was linked by Emmy Noether in the 1920s to the decomposition of ideals in commutative rings, which in turn led Garret Birkhoff in the 1940s to his subdirect representation theorem for general algebras. The Hilbert-Noether-Birkhoff linkage was brought to light in the late 1990s in talks by Bill Lawvere. The aim of this article is to analyze this linkage in the most elementary terms and then, based on our work of the 1980s, to present a general categorical framework for Birkhoff’s theorem.

The paper characterizes the class of subdirectly irreducible algebras satisfying hyperidentities of the variety of De Morgan algebras. Such algebras are called subdirectly irreducible De Morgan quasilattices. The suggested characterization is quite close to that of the classical case of subdirectly irreducible DeMorgan algebras.

Some examples are given of applications of an in nite logical language in universal algebra, in the algebraic geometry of universal algebras, in the theory of implicit operations on algebras, in the Galois theory between automorphisms of universal algebras and its subalgebras of xed points, in the theory of Hamiltonian closure of subalgebras and other areas.

First, we prove two finite algebras are categorically equivalent if and only if the matrix products of their irredundant non-refinable covers are isomorphic. Second, we characterize families of irreducible algebras such that there exists an algebra whose neighbourhoods in an irredundant non-refinable cover are isomorphic to the respective irreducible algebra in the given family. Finally, we exhibit two facts by constructing examples. The first one is that there is a family of irreducible algebras such that there are many algebraic structures whose neighbourhoods in an irredundant non-refinable cover are isomorphic to the respective irreducible algebra in the given family. The second example is an algebra such that the matrix product of an irredundant non-refinable cover is bigger than the given algebra.

We present a complete description of subdirectly irreducible state BL-algebras as well as of subdirectly irreducible state-morphism BL-algebras. In addition, we present a general theory of state-morphism algebras, that is, algebras of general type with state-morphism which is a fixed idempotent endomorphism. We define a diagonal state-morphism algebra and we show that every subdirectly irreducible state-morphism algebra can be embedded into a diagonal one. We describe generators of the varieties of state-morphism algebras, in particular generators of state-morphism BL-algebras, state-morphism MTL-algebras, state-morphism non-associative BL-algebras, and state-morphism pseudo MV-algebras.

In the present paper we will ask for the lattice L(MV Ex ) of subvarieties of the variety defined by the set Ex(MV) of all externally compatible identities valid in the variety MV of all MV-algebras. In particular, we will find all subdirectly irreducible algebras from the classes in the lattice L(MV Ex ) and give syntactical and semantical characterization of the class of algebras defined by P-compatible identities of MV-algebras.

This paper studies absolute retracts in congruence modular varieties of universal algebras. It is shown that every absolute retract with finite dimensional congruence lattice is a product of subdirectly irreducible algebras. Further, every absolute retract in a residually small variety is the product of an abelian algebra and a centerless algebra., 12 pages

We prove some results on the structure of certain classes of integral fusion categories and semisimple Hopf algebras under restrictions on the set of its irreducible degrees., Comment: Revised version. Improved statements of Theorems 4.13 and 7.3

Weak Stone-Ockham algebras are those algebras (L; ∧, ∨, f, ⋆,0,1) of type 〈2,2,1,1,0,0〉, where (L; ∧, ∨, f,0,1) is an Ockham algebra, (L; ∧, ∨, ⋆,0,1) is a weak Stone algebra, and the unary operations f and ⋆ commute. In this paper, we give a complete description of the structure of the lattice of congruences on the subdirectly irreducible algebras.

Here we introduce a subclass of the class of Ockham algebras (L; f) for which L satisfies the property that for every x ∈ L, there exists n ≥ 0 such that fn(x) and fn+1(x) are complementary. We characterize the structure of the lattice of congruences on such an algebra (L; f). We show that the lattice of compact congruences on L is a dual Stone lattice, and in particular, that the lattice Con L of congruences on L is boolean if and only if L is finite boolean. We also show that L is congruence coherent if and only if it is boolean. Finally, we give a sufficient and necessary condition to have the subdirectly irreducible chains.