Let A be an Archimedean f-ring with identity and assume that A is equipped with another multiplication \(*\) so that A is an f-ring with identity u. Obviously, if \(*\) coincides with the original multiplication of A then u is idempotent in A (i.e., \(u^{2}=u\)). Conrad proved that the converse also holds, meaning that, it suffices to have \(u^{2}=u\) to conclude that \(*\) equals the original multiplication

A strongly Fregean algebra is an algebra such that the class of its homomorphic images is Fregean and the variety generated by this algebra is congruence modular. To understand the structure of these algebras we study the prime intervals projectivity relation in the posets of their completely meet irreducible congruences and show that its cosets have the natural structure of a Boolean group. In particular

Let \({\mathcal {A}}\) be a class of algebras with \(I, A \in {\mathcal {A}}\). We interpret the lattice-theoretic “strictly meet irreducible/cover” situation \(B < C\) in lattices of the form \(S_{{\mathcal {A}}}(I,A)\) of all subalgebras of A containing I, where we call such \(B < C\) a minimum proper extension (mpe), and show that this means B is maximal in \(S_{{\mathcal {A}}}(I,A)\) for not containing

For a dynamical system (X, f), the notion of topological transitivity has been studied by some researchers. There are several definitions of this property, and it is part of the folklore of dynamical systems that under some hypotheses, they are equivalent. In this paper, our aim is to introduce and study some properties of topological transitivity in pointfree topology, for which we first need to define

Positive MV-algebras are the subreducts of MV-algebras with respect to the signature \(\{\oplus , \odot , \vee , \wedge , 0, 1\}\). We provide a finite quasi-equational axiomatization for the class of such algebras.

We investigate the representation and complete representation classes for algebras of partial functions with the signature of relative complement and domain restriction. We provide and prove the correctness of a finite equational axiomatisation for the class of algebras representable by partial functions. As a corollary, the same equations axiomatise the algebras representable by injective partial

Based on the previously obtained concrete characterization of the endomorphism semigroups of quasi-acyclic reflexive graphs we prove the relatively elementary definability of the class of such graphs in the class of all semigroups. It will permit us to investigate for such graphs the abstract representation problem for the endomorphism semigroups of graphs and the problem of elementary definability

Using the theory on infinite primes of fields developed by Harrison in [2], the necessary and sufficient conditions are proved for real number fields to be \(O^{*}\)-fields, and many examples of \(O^{*}\)-fields are provided.

We construct non-abelian totally ordered groups of matrices of finite Archimedean rank using the group of o-automorphisms of direct sums of copies of the reals ordered anti-lexicographically. We also prove that each of these o-groups is divisible, and provide, for every \(n>2\), a specific formula to find the n-th root of every element of such group. Finally, we construct an example of a non-commutative

The Mal’tsev product of two varieties of the same similarity type is not in general a variety, because it can fail to be closed under homomorphic images. In the previous paper we provided new sufficient conditions for such a product to be a variety. In this paper we extend that result by weakening the assumptions regarding the two varieties. We also explore the various special cases of our new result

We study the varieties with \(\vec {0}\) and \(\vec {1}\) where factor congruences are definable by existential formulas parameterized by central elements. This continues previous work on equational definability of factor congruences.

We prove that for any act X over a finite semigroup S, the congruence lattice \({{\,\mathrm{Con}\,}}X\) embeds the lattice \({{\,\mathrm{Eq}\,}}M\) of all equivalences of an infinite set M if and only if X is infinite. Equivalently: for an act X over a finite semigroup S, the lattice \({{\,\mathrm{Con}\,}}X\) satisfies a non-trivial identity if and only if X is finite. Similar statements are proved

We introduce the notion of clone algebra (\(\mathsf {CA}\)), intended to found a one-sorted, purely algebraic theory of clones. \(\mathsf {CA}\)s are defined by identities and thus form a variety in the sense of universal algebra. The most natural \(\mathsf {CA}\)s, the ones the axioms are intended to characterise, are algebras of functions, called functional clone algebras (\(\mathsf {FCA}\)). The

In this paper we study the cancellation law for the tensor product in the category \(\mathsf {Sup}\) of complete lattices and join-preserving maps. First, we investigate the tensor product of generalized power-set lattices. Based on which, we prove that the cancellation law for the tensor product has a close relation to that for the cartesian product of posets, and give a class of complete lattices

We study residuated homomorphisms (r-morphisms) and their adjoints, the so-called localizations (or l-morphisms), between implicative semilattices, because these objects may be characterized as semilattices whose unary meet operations have adjoints. Since left resp. right adjoint maps are the residuated resp. residual maps (having the property that preimages of principal downsets resp. upsets are again

In a paper published in 2015, we introduced TiRS graphs and TiRS frames to create a new natural setting for duals of canonical extensions of lattices. Here, we firstly introduce morphisms of TiRS structures and put our correspondence between TiRS graphs and TiRS frames into a full categorical framework. We then answer Problem 2 from our 2015 paper by characterising the perfect lattices that are dual

We extend the concept of “almost indiscernible theory” introduced by Pillay and Sklinos in 2015 (which was itself a modernization and expansion of Baldwin and Shelah from 1983), to uncountable languages and uncountable parameter sequences. Roughly speaking a theory \(T\) is almost indiscernible if some saturated model is in the algebraic closure of an indiscernible set of sequences. We show that such

A new protomodular analog of the classical criterion for the existence of a group term in the algebraic theory of a variety of universal algebras is given. To this end, the notion of a right-cancellable protomodular algebra is introduced. The translation group functor from the category of right-cancellable algebras of a protomodular variety to the category of groups is constructed. It is proved that

Call a directed partially ordered cancellative divisibility monoid M a Riesz monoid if for all \(x,y_{1},y_{2}\ge 0\) in M, \(x\le y_{1}+y_{2}\Rightarrow x=x_{1}+x_{2}\) where \(0\le x_{i}\le y_{i}\). We explore the necessary and sufficient conditions under which a Riesz monoid M with \(M^{+}=\{x\ge 0\mid x\in M\}=M\) generates a Riesz group and indicate some applications. We call a directed p.o. monoid

The main purpose of this paper is a wide generalization of one of the results abstract algebraic geometry begins with, namely of the fact that the prime spectrum \({\mathrm {Spec}}(R)\) of a unital commutative ring R is always a spectral (= coherent) topological space. In this generalization, which includes several other known ones, the role of ideals of R is played by elements of an abstract complete

Mitschke showed that a variety with an m-ary near-unanimity term has Jónsson terms \(t_0, \dots , t _{2m-4} \) witnessing congruence distributivity. We show that Mitschke’s result is sharp. We also evaluate the best possible number of Day terms witnessing congruence modularity. More generally, we characterize exactly the best bounds for many congruence identities satisfied by varieties with an m-ary

In the author’s recent paper, the free left (right) n-trinilpotent trioid was constructed. Our aim is to characterize the least left (right) n-trinilpotent congruence on the free trioid.

We study the computational complexity of the satisfiability problem and the complement of the equivalence problem for complemented (orthocomplemented) modular lattices L and classes thereof. Concerning a simple L of finite height, \(\mathcal {NP}\)-hardness is shown for both problems. Moreover, both problems are shown to be polynomial-time equivalent to the same feasibility problem over the division

We abstract the notion of fraction-density of f-rings (introduced by Anthony Hager and Jorge Martínez) to algebraic frames. We say an algebraic frame with the finite intersection property on compact elements is fraction-dense if each of its polars is a polar of a compact element. This turns out to be a “conservative” extension of the fraction-density property in the sense that a reduced f-ring is fraction-dense

By several postulates we introduce a new class of algebraic lattices, in which a main role is played by so called normal elements. A model of these lattices are weak-congruence lattices of groups, so that normal elements correspond to normal subgroups of subgroups. We prove that in this framework many basic structural properties of groups turn out to be lattice-theoretic. Consequently, we give necessary

We construct the free products of arbitrary digroups, and thus we solve an open problem of Zhuchok.

The system \({\mathcal {S}}_c(L)\) consisting of joins of closed sublocales of a locale L is known to be a frame, and for L subfit it coincides with the Booleanization \({\mathcal {S}}_b(L)\) of the coframe of sublocales of L. In this paper, we study \({\mathcal {S}}_b(L)\) for a general locale L. We show that \({\mathcal {S}}_c(L)\) is always a subframe of \({\mathcal {S}}_b(L)\). Moreover, if X is

We provide a simple proof of a recent result of Morton and van Alten that the canonical extension of a bounded distributive lattice is its free completely distributive extension. We show that this can be used to easily obtain a number of results, both known and new.

Abstract We show that Lemma 4.4 and Theorem 4.5 in [Zhang, X., Laan, V., Injective hulls for ordered algebras, Algebra Universalis, 76 (2016), 339–349] are incorrect. These results can be corrected by replacing unary polynomial functions by linear functions.

This paper considers the complexity of the problem of determining whether a finite idempotent algebra generates a variety which satisfies various strong Mal’tsev conditions. In particular we show that if the strong Mal’tsev condition under consideration implies the existence of an edge term, then deciding whether a finite idempotent algebra generates a variety which satisfies the Mal’tsev condition

If \({\mathbf {G}}=(G;\cdot )\) is a semigroup, I is arbitrary set and \(\lambda ,\rho :I\rightarrow I\) are mappings satisfying the equalities \(\lambda \lambda =\lambda \), \(\rho \rho =\rho \) and \(\lambda \rho =\rho \lambda \) then we define the semigroup \((G^I,\times )\) where \((x\times y)(i) := x( \lambda i)\cdot y (\rho i)\). This construction gives rise to four covariant and two contravariant

We investigate the finitary functions from a finite product of finite fields \(\prod _{j =1}^m\mathbb {F}_{q_j} = {\mathbb K}\) to a finite product of finite fields \(\prod _{i =1}^n\mathbb {F}_{p_i} = {\mathbb {F}}\), where \(|{\mathbb K}|\) and \(|{\mathbb {F}}|\) are coprime. An \(({\mathbb {F}},{\mathbb K})\)-linearly closed clonoid is a subset of these functions which is closed under composition

A frame is a complete lattice in which the meet distributes over arbitrary joins. Let \(\tau \) be a subframe of a frame L such that every element of \(\tau \) has a complement in L, then \((L, \tau )\), briefly \(L_{ \tau }\), is said to be a topoframe. Let \({\mathcal {R}}L_\tau \) be the ring of real-continuous functions on a topoframe \(L_{ \tau }\). We define P-topoframes and show that \(L_{\tau

A subresiduated lattice is a pair (A, D), where A is a bounded distributive lattice, D is a bounded sublattice of A and for every \(a,b\in A\) there is \(c\in D\) such that for all \(d\in D\), \(d\wedge a\le b\) if and only if \(d\le c\). This c is denoted by \(a\rightarrow b\). This pair can be regarded as an algebra \(\left\) of type (2, 2, 2, 0, 0) where \(D=\{a\in A\mid 1\rightarrow a=a\}\). The

An algebra \(\mathbf{A}\) is called a perfect extension of its subalgebra \(\mathbf{B}\) if every congruence of \(\mathbf{B}\) has a unique extension to \(\mathbf{A}\). This terminology was used by Blyth and Varlet [1994]. In the case of lattices, this concept was described by Grätzer and Wehrung [1999] by saying that \(\mathbf{A}\) is a congruence-preserving extension of \(\mathbf{B}\). Not many investigations

We characterize commutative idempotent involutive residuated lattices as disjoint unions of Boolean algebras arranged over a distributive lattice. We use this description to introduce a new construction, called gluing, that allows us to build new members of this variety from other ones. In particular, all finite members can be constructed in this way from Boolean algebras. Finally, we apply our construction

In this paper we investigate measures over bounded lattices, extending and giving a unifying treatment to previous works. In particular, we prove that the measures of an arbitrary bounded lattice can be represented as measures over a suitably chosen Boolean lattice. Using techniques from algebraic geometry, we also prove that given a bounded lattice X there exists a scheme \(\mathcal {X}\) such that

We present a functorial construction which, starting from a congruence \(\alpha \) of finite index in an algebra \(\mathbf {A}\), yields a new algebra \(\mathbf {C}\) with the following properties: the congruence lattice of \(\mathbf {C}\) is isomorphic to the interval of congruences between 0 and \(\alpha \) on \(\mathbf {A}\), this isomorphism preserves higher commutators and TCT types, and \(\mathbf

In the theory of semigroups there exists a construction of some semilattices from a special family of semigroups. It is the so-called ‘strong semilattice of semigroups’. Modifying this idea we can present a new construction method for arbitrary pseudocomplemented semilattice (= PCS) L using ‘full triples’. This construction centers around the classical Glivenko-Frink congruence \(\Gamma (L)\). This

We investigate the lattice of clones that are generated by a set of functions that are induced on a finite field \({\mathbb {F}}\) by monomials. We study the atoms and coatoms of this lattice and investigate whether this lattice contains infinite ascending chains, or infinite descending chains, or infinite antichains.We give a connection between the lattice of these clones and semi-affine algebras

The collection of hereditary classes of modules over an arbitrary ring R is a pseudocomplemented complete big lattice. The elements of its skeleton are precisely the natural classes of R-modules. In this paper we extend some results about hereditary classes in R-Mod to the category \(\mathcal {L}_\mathcal {M}\) of linear modular lattices, which has as objects all modular complete lattices and as morphisms

A join-semilattice L with top is said to be conjunctive if every principal ideal is an intersection of maximal ideals. (This is equivalent to a first-order condition in the language of semilattices.) In this paper, we explore the consequences of the conjunctivity hypothesis for L, and we define and study a related property, called “ideal conjunctivity,” which is applicable to join-semilattices without

The Galois correspondence \({{\mathbf {G}}}\) between sets of logical formulas of the first order language \(\mathbf{L}\) over a signature L and type definable sets over an \(\mathbf{L}\)-structure A on the set \({{\mathbf {A}}}\) is extensively studied in the literature. The Stone topology is successfully applied in model theory. We investigate basic properties of the pointwise convergence topologies

In this paper we use the theory of central elements in order to provide a characterization for coextensive varieties. In particular, if a variety is of finite type, congruence-permutable and its class of directly indecomposable members is universal, then the variety is coextensive if and only if it is a variety of shells.

It is proven in this note that if non-empty posets A, B, C, and D satisfy \(A^C\cong B^D\) where C, D, and \(A^C\) are finite and connected, then there exist posets E, X, Y, and Z such that \(A\cong E^X\), \(B\cong E^Y\), \(C\cong Y\times Z\), and \(D\cong X\times Z\). This solves a problem posed by Jónsson and McKenzie in 1982.

We investigate the computational complexity of the problem of deciding if an algebra homomorphism can be factored through an intermediate algebra. Specifically, we fix an algebraic language, \(\mathcal L\), and take as input an algebra homomorphism \(f:X\rightarrow Z\) between two finite \(\mathcal L\)-algebras X and Z, along with an intermediate finite \(\mathcal L\)-algebra Y. The decision problem

An algebra-valued model of set theory is called loyal to its algebra if the model and its algebra have the same propositional logic; it is called faithful if all elements of the algebra are truth values of a sentence of the language of set theory in the model. We observe that non-trivial automorphisms of the algebra result in models that are not faithful and apply this to construct three classes of

We provide a generalization of Mundici’s equivalence between unital Abelian lattice-ordered groups and MV-algebras: the category of unital commutative lattice-ordered monoids is equivalent to the category of MV-monoidal algebras. Roughly speaking, unital commutative lattice-ordered monoids are unital Abelian lattice-ordered groups without the unary operation \(x \mapsto -x\). The primitive operations

Let \(\mathbb {S}\) be the commutative and idempotent semiring with additive identity \(\mathbf {0}\) and multiplicative identity \(\mathbf {1}\). The tropical semiring \(\mathbb {T}\) and the Boolean semiring \(\mathbb {B}\) are common important examples of such semirings. Let \(UT_{n}(\mathbb {S})\) be the semigroup of all \(n\times n\) upper triangular matrices over \(\mathbb {S}\), both \(UT^{\pm