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Abstract: Abstract 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 and provide a number of examples of its application. PubDate: 2022-05-21

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Abstract: Abstract Suppose p(x, y, z) and q(x, y, z) are terms. If there is a common “ancestor” term \(s(z_{1},z_{2},z_{3},z_{4})\) specializing to p and q through identifying some variables $$\begin{aligned} p(x,y,z)&\approx s(x,y,z,z)\\ q(x,y,z)&\approx s(x,x,y,z), \end{aligned}$$ then the equation $$\begin{aligned} p(x,x,z)\approx q(x,z,z) \end{aligned}$$ is a trivial consequence. In this note we show that for lattice terms, and more generally for terms of lattice-ordered algebras, a converse is true, too. Given terms p, q, and an equation where \(\{u_{1},\ldots ,u_{m}\}=\{v_{1},\ldots ,v_{n}\},\) there is always an “ancestor term” \(s(z_{1},\ldots ,z_{r})\) such that \(p(x_{1},\ldots ,x_{m})\) and \(q(y_{1},\ldots ,y_{n})\) arise as substitution instances of s, whose unification results in the original equation ( \(*\) ). In category theoretic terms the above proposition, when restricted to lattices, has a much more concise formulation:Free-lattice functors weakly preserve pullbacks of epis. Finally, we show that weak preservation is all that we can hope for. We prove that for an arbitrary idempotent variety \({{\mathcal {V}}}\) the free-algebra functor \(F_{{\mathcal {V}}}\) will not preserve pullbacks of epis unless \({{\mathcal {V}}}\) is trivial (satisfying \(x\approx y\) ) or \({{\mathcal {V}}}\) contains the “variety of sets” (where all operations are implemented as projections). PubDate: 2022-04-23

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Abstract: Abstract 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. PubDate: 2022-04-22

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Abstract: Abstract A nucleus on a meet-semilattice A is a closure operation that preserves binary meets. The nuclei form a semilattice \(\mathrm{N }A\) that is isomorphic to the system \({\mathcal {N}}A\) of all nuclear ranges, ordered by dual inclusion. The nuclear ranges are those closure ranges which are total subalgebras (l-ideals). Nuclei have been studied intensively in the case of complete Heyting algebras. We extend, as far as possible, results on nuclei and their ranges to the non-complete setting of implicative semilattices (whose unary meet translations have adjoints). A central tool are so-called r-morphisms, that is, residuated semilattice homomorphisms, and their adjoints, the l-morphisms. Such morphisms transport nuclear ranges and preserve implicativity. Certain completeness properties are necessary and sufficient for the existence of a least nucleus above a prenucleus or of a greatest nucleus below a weak nucleus. As in pointfree topology, of great importance for structural investigations are three specific kinds of l-ideals, called basic open, boolean and basic closed. PubDate: 2022-04-12

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Abstract: Abstract We extend the validity of Kiss’s characterization of “ \([\alpha ,\beta ]=0\) ” from congruence modular varieties to varieties with a difference term. This fixes a recently discovered gap in our paper Kearnes et al. (Trans Am Math Soc 368:2115–2143, 2016). We also prove some related properties of Kiss terms in varieties with a difference term. PubDate: 2022-04-06

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Abstract: Abstract 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 for an act with zero over a completely 0-simple semigroup \({\mathcal {M}}^0(G,I,\Lambda ,P)\) where \( G , I <\infty \) . We construct examples that show that the assumption \( G , I <\infty \) is essential. PubDate: 2022-04-05

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Abstract: Abstract 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 universe of a \(\mathsf {FCA}\) , called \(\omega \) -clone, is a set of infinitary operations on a given set, containing the projections and closed under finitary compositions. The main result of this paper is the general representation theorem, where it is shown that every \(\mathsf {CA}\) is isomorphic to a \(\mathsf {FCA}\) and that the variety \(\mathsf {CA}\) is generated by the class of finite-dimensional \(\mathsf {CA}\) s. This implies that every \(\omega \) -clone is algebraically generated by a suitable family of clones by using direct products, subalgebras and homomorphic images. We conclude the paper with two applications. In the first one, we use clone algebras to give an answer to a classical question about the lattices of equational theories. The second application is to the study of the category of all varieties of algebras. PubDate: 2022-03-20

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Abstract: Abstract 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 which do not satisfy the cancellation law for the tensor product. Then, we also investigate the cancellation law for particular subclasses of complete lattices. PubDate: 2022-03-20

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Abstract: Abstract 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 to TiRS frames (and hence TiRS graphs). We introduce a new subclass of perfect lattices called PTi lattices and show that the canonical extensions of lattices are PTi lattices, and so are ‘more’ than just perfect lattices. We illustrate the correspondences between classes of our newly-described PTi lattices and classes of TiRS graphs by examples. We conclude by outlining a direction for future research. PubDate: 2022-03-19

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Abstract: 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 such), one may not only regard the l-morphisms as abstract continuous maps in a pointfree framework (as familiar in the complete case), but also characterize them by concrete closure-theoretical continuity properties. These concepts apply to locales (frames, complete Heyting lattices) and provide generalizations of continuous and open maps between spaces to an algebraic (not necessarily complete) pointfree setting. PubDate: 2022-03-19

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Abstract: Abstract 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 theory \(T\) is nonmultidimensional, superstable, and stable in all cardinals \(\ge T \) . We prove a structure theorem for sufficiently large \( a \) -models \(M\) , which states that over a suitable base, \(M\) is in the algebraic closure of an independent set of realizations of weight one types (in possibly infinitely many variables). We also explore further the saturated free algebras of Baldwin and Shelah in both the countable and uncountable context. We study in particular theories and varieties of \(R\) -modules, characterizing those rings \( R \) for which the free \( R \) -module on \( \left R\right ^{+} \) generators is saturated, and pointing out a counterexample to a conjecture by Pillay and Sklinos. PubDate: 2022-02-01 DOI: 10.1007/s00012-021-00766-x

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Abstract: Abstract 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 the algebraic theory of a variety of universal algebras contains a group term if and only if it contains protomodular terms with respect to which all algebras from the variety are right-cancellable. Moreover, the right-cancellable algebras from the simplest protomodular varieties are characterized as sets with principal group actions as well as groups with simple additional structures. PubDate: 2022-01-29 DOI: 10.1007/s00012-021-00747-0

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Abstract: Abstract 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 M \(\Pi \) -pre-Riesz if \( M^{+}=M\) and for all \(x_{1},x_{2}, \dots ,x_{n}\in M\) , \({{\,\mathrm{glb}\,}}(x_{1},x_{2},\dots ,x_{n})=0\) or there is \(r\in \Pi \) such that \(0<r\le x_{1},x_{2},\dots ,x_{n},\) for some subset \(\Pi \) of M. We explore examples of \(\Pi \) -pre-Riesz monoids of \(*\) -ideals of different types. We show for instance that if M is the monoid of nonzero (integral) ideals of a Noetherian domain D and \(\Pi \) the set of invertible ideals, M is \(\Pi \) -pre-Riesz if and only D is a Dedekind domain. We also study factorization in pre-Riesz monoids of a certain type and link it with factorization theory of ideals in an integral domain. PubDate: 2022-01-16 DOI: 10.1007/s00012-021-00765-y

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Abstract: 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 lattice L equipped with a binary multiplication with \(xy\leqslant x\wedge y\) for all \(x,y\in L\) . In fact when no further conditions on L are required, the resulting space can be and is only shown to be sober, and we discuss further conditions sufficient to make it spectral. This discussion involves establishing various comparison theorems on so-called prime, radical, solvable, and locally solvable elements of L; we also make short additional remarks on semiprime elements. We consider categorical and universal-algebraic applications involving general theory of commutators, and an application to ideals in what we call the commutative world. The cases of groups and of non-commutative rings are briefly considered separately. PubDate: 2022-01-16 DOI: 10.1007/s00012-021-00764-z

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Abstract: Abstract 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 near-unanimity term. Finally we present some simple observations about terms with just one “dissenter”, a generalization of a minority term. PubDate: 2022-01-12 DOI: 10.1007/s00012-021-00762-1

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Abstract: Abstract 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. PubDate: 2021-12-31 DOI: 10.1007/s00012-021-00758-x

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Abstract: Abstract 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 ring D whenever L is the subspace lattice of a D-vector space of finite dimension at least 3. Considering the class of all finite dimensional Hilbert spaces, the equivalence problem for the class of subspace ortholattices is shown to be polynomial-time equivalent to that for the class of endomorphism \(*\) -rings with pseudo-inversion; moreover, we derive completeness for the complement of the Boolean part of the nondeterministic Blum-Shub-Smale model of real computation without constants. This result extends to the additive category of finite dimensional Hilbert spaces, enriched by adjunction and pseudo-inversion. PubDate: 2021-12-31 DOI: 10.1007/s00012-021-00760-3

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Abstract: 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 precisely when its frame of radical ideals is fraction-dense. We characterize these frames and study properties of some other types of algebraic frames that arise naturally in the characterizations of the fraction-dense ones. PubDate: 2021-12-31 DOI: 10.1007/s00012-021-00763-0

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Abstract: Abstract 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 and sufficient conditions under which a group is Hamiltonian, Dedekind, abelian, solvable, supersolvable, metabelian, finite nilpotent. These conditions are given as lattice-theoretic properties of a lattice with normal elements. PubDate: 2021-11-29 DOI: 10.1007/s00012-021-00759-w

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Abstract: Abstract We construct the free products of arbitrary digroups, and thus we solve an open problem of Zhuchok. PubDate: 2021-11-29 DOI: 10.1007/s00012-021-00761-2