Fully invariant and verbal congruence relations

A congruence relation θ on an algebra A is fully invariant if every endomorphism of A preserves θ. A congruence θ is verbal if there exists a variety ${\mathcal{V}}$ such that θ is the least congruence of A such that ${{\bf A}/\theta \in \mathcal{V}}$ . Every verbal congruence relation is known to be fully invariant. This paper investigates fully invariant congruence relations that are verbal, algebras whose fully invariant congruences are verbal, and varieties for which every fully invariant congruence in every algebra in the variety is verbal.     

A uniform Birkhoff theorem

Birkhoff’s HSP theorem characterizes the classes of models of algebraic theories as those being closed with respect to homomorphic images, subalgebras, and products. In particular, it implies that an algebra B satisfies all equations that hold in an algebra A of the same signature if and only if B is a homomorphic image of a subalgebra of a (possibly infinite) direct power of A. The former statement is equivalent to the existence of a natural map sending term functions of the algebra A to those of B—the natural clone homomorphism. The study of continuity properties of natural clone homomorphisms has been initiated recently by Bodirsky and Pinsker for locally oligomorphic algebras.Revisiting the argument of Bodirsky and Pinsker, we show that for any algebra B in the variety generated by an algebra A, the induced natural clone homomorphism is uniformly continuous if and only if every finitely generated subalgebra of B is a homomorphic image of a subalgebra of a finite power of A. Based on this observation, we study the question as to when Cauchy continuity of natural clone homomorphisms implies uniform continuity. We introduce the class of almost locally finite algebras, which encompasses all locally oligomorphic as well as all locally finite algebras, and show that, in case A is almost locally finite, then the considered natural homomorphism is uniformly continuous if (and only if) it is Cauchy-continuous. In particular, this provides a locally finite counterpart of the result by Bodirsky and Pinsker. Along the way, we also discuss some peculiarities of oligomorphic permutation groups on uncountable sets.     

Finitely Related Clones and Algebras with Cube Terms

Aichinger et al. (2011) have proved that every finite algebra with a cube-term (equivalently, with a parallelogram-term; equivalently, having few subpowers) is finitely related. Thus finite algebras with cube terms are inherently finitely related??every expansion of the algebra by adding more operations is finitely related. In this paper, we show that conversely, if A is a finite idempotent algebra and every idempotent expansion of A is finitely related, then A has a cube-term. We present further characterizations of the class of finite idempotent algebras having cube-terms, one of which yields, for idempotent algebras with finitely many basic operations and a fixed finite universe A, a polynomial-time algorithm for determining if the algebra has a cube-term. We also determine the maximal non-finitely related idempotent clones over A. The number of these clones is finite.     

Automata all of whose congruences are inner

A congruence on an automaton A is called inner if it is the kernel of a certain endomorphism on A. We propose a characterization of automata, all of whose congruences are inner.     

Subdirect products of hereditary congruence lattices

A congruence lattice L of an algebra A is called power-hereditary if every 0-1 sublattice of Lⁿ is the congruence lattice of an algebra on Aⁿ for all positive integers n. Let A and B be finite algebras. We prove

Nilpotent and solvable radicals in locally finite congruence modular varieties

The Loewy rank of a modular latticeL of finite height is defined as the leastn for which there exista ₀=0t, < ... r=1 inL such that each interval I[a_i, a_i+1] is a complemented lattice. In this paper, a generalized notion of Loewy rank is applied to obtain new results in the commutator theory of locally finite congruence modular varieties. LetV be a finitely generated congruence modular variety. We prove that every algebra inV has a largest nilpotent congruence and a largest solvable congruence. Moreover, there exist first order formulas which define these special congruences in every algebra ofV.     

Boolean topological graphs of semigroups: the lack of first-order axiomatization

The graph of an algebra A is the relational structure G(A) in which the relations are the graphs of the basic operations of A. For a class ?? of algebras let G(??)={G(A)∣A∈??}. Assume that ?? is a class of semigroups possessing a nontrivial member with a neutral element and let ? be the universal Horn class generated by G(??). We prove that the Boolean core of ?, i.e., the topological prevariety generated by finite members of ? equipped with the discrete topology, does not admit a first-order axiomatization relative to the class of all Boolean topological structures in the language of ?. We derive analogous results when ?? is a class of monoids or groups with a nontrivial member.     

The commutator in equivalential algebras and Fregean varieties

A class \({\mathcal {K}}\) of algebras with a distinguished constant term 0 is called Fregean if congruences of algebras in \({\mathcal {K}}\) are uniquely determined by their 0-cosets and Θ _A (0, a) = Θ _A (0, b) implies a = b for all \({a, b \in {\bf A} \in \mathcal {K}}\) . The structure of Fregean varieties was investigated in a paper by P. Idziak, K. S?omczyńska, and A. Wroński. In particular, it was shown there that every congruence permutable Fregean variety consists of algebras that are expansions of equivalential algebras, i.e., algebras that form an algebraization of the purely equivalential fragment of the intuitionistic propositional logic. In this paper we give a full characterization of the commutator for equivalential algebras and solvable Fregean varieties. In particular, we show that in a solvable algebra from a Fregean variety, the commutator coincides with the commutator of its purely equivalential reduct. Moreover, an intrinsic characterization of the commutator in this setting is given.     

Reflexive relations on algebras with Boolean lattice reducts

For any algebra A, let Ref(A) be the algebra of compatible reflexive binary relations on A under intersection, composition, and converse with the universal and identity relations as constants. We characterize all Ref(A) where A is a finite algebra with a Boolean lattice reduct.     

Meet-irreducible congruence lattices

The system of all congruences of an algebra (A, F) forms a lattice, denoted \({{\mathrm{Con}}}(A, F)\). Further, the system of all congruence lattices of all algebras with the base set A forms a lattice \(\mathcal {E}_A\). We deal with meet-irreducibility in \(\mathcal {E}_A\) for a given finite set A. All meet-irreducible elements of \(\mathcal {E}_A\) are congruence lattices of monounary algebras. Some types of meet-irreducible congruence lattices were described in Jakubíková-Studenovská et al. (2017). In this paper, we prove necessary and sufficient conditions under which \({{\mathrm{Con}}}(A, f)\) is meet-irreducible in the case when (A, f) is an algebra with short tails (i.e., f(x) is cyclic for each \(x \in A\)) and in the case when (A, f) is an algebra with small cycles (every cycle contains at most two elements).     

Lattice subordinations and Priestley duality

There is a well-known correspondence between Heyting algebras and S4-algebras. Our aim is to extend this correspondence to distributive lattices by defining analogues of S4-algebras for them. For this purpose, we introduce binary relations on Boolean algebras that resemble de Vries proximities. We term such binary relations lattice subordinations. We show that the correspondence between Heyting algebras and S4-algebras extends naturally to distributive lattices and Boolean algebras with a lattice subordination. We also introduce Heyting lattice subordinations and prove that the category of Boolean algebras with a Heyting lattice subordination is isomorphic to the category of S4-algebras, thus obtaining the correspondence between Heyting algebras and S4-algebras as a particular case of our approach. In addition, we provide a uniform approach to dualities for these classes of algebras. Namely, we generalize Priestley spaces to quasi-ordered Priestley spaces and show that lattice subordinations on a Boolean algebra B correspond to Priestley quasiorders on the Stone space of B. This results in a duality between the category of Boolean algebras with a lattice subordination and the category of quasi-ordered Priestley spaces that restricts to Priestley duality for distributive lattices. We also prove that Heyting lattice subordinations on B correspond to Esakia quasi-orders on the Stone space of B. This yields Esakia duality for S4-algebras, which restricts to Esakia duality for Heyting algebras.     

Semiconic idempotent residuated structures

An idempotent residuated po-monoid is semiconic if it is a subdirect product of algebras in which the monoid identity is comparable with all other elements. It is proved that the quasivariety SCIP of all semiconic idempotent commutative residuated po-monoids is locally finite. The lattice-ordered members of this class form avariety SCIL, which is not locally finite, but it is proved that SCIL has the finite embeddability property (FEP). More generally, for every relative subvariety K of SCIP, the lattice-ordered members of K have the FEP. This gives a unified explanation of the strong finite model property for a range of logical systems. It is also proved that SCIL has continuously many semisimple subvarieties, and that the involutive algebras in SCIL are subdirect products of chains.     

Triangular irreducibility of congruences in quasivarieties

Certain forms of irreducibility as well as of equational definability of relative congruences in quasivarieties are investigated. For any integer ${m \geqslant 3}$ and a quasivariety Q, the notion of an m-triangularily meet-irreducible Q-congruence in the algebras of Q is defined. In Section 2, some characterizations of finitely generated quasivarieties involving this notion are provided. Section 3 deals with quasivarieties with equationally definable m-triangular meets of relatively principal congruences. References to finitely based quasivarieties and varieties are discussed.     

Monadic MV-algebras I: a study of subvarieties

In this paper, we study and classify some important subvarieties of the variety of monadic MV-algebras. We introduce the notion of width of a monadic MV-algebra and we prove that the equational class of monadic MV-algebras of finite width k is generated by the monadic MV-algebra [0, 1] ^k . We describe completely the lattice of subvarieties of the subvariety ${\mathcal{V}([{\bf 0}, {\bf 1}]^k)}$ generated by [0, 1] ^k . We prove that the subvariety generated by a subdirectly irreducible monadic MV-algebra of finite width depends on the order and rank of ?A, the partition associated to A of the set of coatoms of the boolean subalgebra B(A) of its complemented elements, and the width of the algebra. We also give an equational basis for each proper subvariety in ${\mathcal{V}([{\bf 0}, {\bf 1}]^k)}$ . Finally, we give some results about subvarieties of infinite width.     

Sequences of Commutator Operations

Given the congruence lattice ${{\mathbb{L}}}$ of a finite algebra A with a Mal’cev term, we look for those sequences of operations on ${{\mathbb{L}}}$ that are sequences of higher commutator operations of expansions of A. The properties of higher commutators proved so far delimit the number of such sequences: the number is always at most countably infinite; if it is infinite, then ${{\mathbb{L}}}$ is the union of two proper subintervals with nonempty intersection.     

On the epicomplete monoreflection of an archimedean lattice-ordered group

In a category C an object it G is epicomplete if the only epic monics out of G are isomorphisms, epic or monic meant in the categorical sense of right or left cancellable. For each of the categories Arch: archimedean ?-groups with ?-homomorphisms, and its companion category W: Arch-objects with distinguished weak unit and unit-preserving ?-homomorphisms, (and for the corresponding categories of vector lattices) epicompleteness has been characterized as divisible and conditionally and laterally σ-complete, and it has been shown to be monoreflective. Denote the reflecting functors by β and β ^W , respectively. What are they? For W the Yosida representation has been used to realize β ^W A as a certain quotient of B (Y A), the Baire functions on the Yosida space of A. For Arch, very little has been known. Here we give a general representation theorem, Theorem A, for β G as a certain subdirect product of W-epicomplete objects derived from G. That result, some W-theory, and the relation between epicity and relative uniform density are then employed to show Theorem B: β C _K (Y)=B _L (Y), where C _K (Y)is the ?-group of continuous functions on Y with compact support and B _L (Y) is the ?-group of Baire functions on Y having Lindelöf cozero sets.     

MacNeille completion and profinite completion can coincide on finitely generated modal algebras

Following Bezhanishvili and Vosmaer, we confirm a conjecture of Yde Venema by piecing together results from various authors. Specifically, we show that if ${\mathbb{A}}$ is a residually finite, finitely generated modal algebra such that HSP( ${\mathbb{A}}$ ) has equationally definable principal congruences, then the profinite completion of ${\mathbb{A}}$ is isomorphic to its MacNeille completion, and ? is smooth. Specific examples of such modal algebras are the free K4-algebra and the free PDL-algebra.