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Abstract: Abstract Motivated by the classical work of Halmos on functional monadic Boolean algebras, we derive three basic sup-semilattice constructions, among other things, the so-called powersets and powerset operators. Such constructions are extremely useful and can be found in almost all branches of modern mathematics, including algebra, logic, and topology. Our three constructions give rise to four covariant and two contravariant functors and constitute three adjoint situations we illustrate in simple examples. PubDate: 2022-11-01

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Abstract: Abstract We explore a pointfree theory of bitopological spaces (that is, sets equipped with two topologies). In particular, here we regard finitary biframes as duals of bitopological spaces. In particular for a finitary biframe \(\mathcal {L}\) the ordered collection of all its pointfree bisubspaces (i.e. its biquotients) is studied. It is shown that this collection is bitopological in three meaningful ways. In particular it is shown that, apart from the assembly of a finitary biframe \(\mathcal {L}\) , there are two other structures \(\mathsf {A}_{cf}(\mathcal {L})\) and \(\mathsf {A}_{\pm }(\mathcal {L})\) , which both have the same main component as \(\mathsf {A}(\mathcal {L})\) . The main component of both \(\mathsf {A}_{cf}(\mathcal {L})\) and \(\mathsf {A}_{\pm }(\mathcal {L})\) is the ordered collection of all biquotients of \(\mathcal {L}\) . The structure \(\mathsf {A}_{cf}(\mathcal {L})\) being a biframe shows that the collection of all biquotients is generated by the frame of the patch-closed biquotients together with that of the patch-fitted ones. The structure \(\mathsf {A}_{\pm }(\mathcal {L})\) being a biframe shows the collection of all biquotients is generated by the frame of the positive biquotients together with that of the negative ones. Notions of fitness and subfitness for finitary biframes are introduced, and it is shown that the analogues of two characterization theorems for these axioms hold. A spatial, bitopological version of these theorems is proven, in which finitary biframes whose spectrum is pairwise \(T_1\) are characterized, among other things in terms of the spectrum \(\mathsf {bpt}(\mathsf {A}_{cf}(\mathcal {L}))\) . PubDate: 2022-10-08

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Abstract: Abstract This paper concerns uniform continuity of real-valued functions on a (pre-)uniform frame. The aim of the paper is to characterize uniform continuity of such frame homomorphisms in terms of a farness relation between elements in the frame, and then to derive from it a separation and an extension theorem for real-valued uniform maps on uniform frames. The approach, purely order-theoretic, uses properties of the Galois maps associated with the farness relation. As a byproduct, we identify sufficient conditions under which a (continuous) scale in a frame with a preuniformity generates a real-valued uniform map. PubDate: 2022-10-03

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Abstract: Abstract The existing topological representation of an orthocomplemented lattice via the clopen orthoregular subsets of a Stone space depends upon Alexander’s Subbase Theorem, which asserts that a topological space X is compact if every subbasic open cover of X admits of a finite subcover. This is an easy consequence of the Ultrafilter Theorem—whose proof depends upon Zorn’s Lemma, which is well known to be equivalent to the Axiom of Choice. Within this work, we give a choice-free topological representation of orthocomplemented lattices by means of a special subclass of spectral spaces; choice-free in the sense that our representation avoids use of Alexander’s Subbase Theorem, along with its associated nonconstructive choice principles. We then introduce a new subclass of spectral spaces which we call upper Vietoris orthospaces in order to characterize up to homeomorphism (and isomorphism with respect to their orthospace reducts) the spectral spaces of proper lattice filters used in our representation. It is then shown how our constructions give rise to a choice-free dual equivalence of categories between the category of orthocomplemented lattices and the dual category of upper Vietoris orthospaces. Our duality combines Bezhanishvili and Holliday’s choice-free spectral space approach to Stone duality for Boolean algebras with Goldblatt and Bimbó’s choice-dependent orthospace approach to Stone duality for orthocomplemented lattices. PubDate: 2022-08-07 DOI: 10.1007/s00012-022-00789-y

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Abstract: Abstract A well-known result in quasigroup theory says that an associative quasigroup is a group, i.e. in quasigroups, associativity forces the existence of an identity element. The converse is, of course, far from true, as there are many, many non-associative loops. However, a remarkable theorem due to David Mumford and C.P. Ramanujam says that any projective variety having a binary morphism admitting a two-sided identity must be a group. Motivated by this result, we define a universal algebra (A; F) to be an MR-algebra if whenever a binary term function m(x, y) in the algebra admits a two-sided identity, then the reduct (A; m(x, y)) must be associative. Here we give some non-trivial varieties of quasigroups, groups, rings, fields and lattices which are MR-algebras. For example, every MR-quasigroup must be isotopic to a group, MR-groups are exactly the nilpotent groups of class 2, while commutative rings and complemented lattices are MR-algebras if and only if they are Boolean. PubDate: 2022-08-07 DOI: 10.1007/s00012-022-00790-5

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Abstract: Abstract 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 on A. The main purpose of this paper is to extend this result as follows. Let A be a (not necessary unital) Archimedean f-ring and B be an \(\ell \) -subgroup of the underlaying \(\ell \) -group of A. We will prove that if B is an f-ring with identity u, then the equality \(u^{2}=u\) is a necessary and sufficient condition for B to be an f-subring of A. As a key step in the proof of this generalization, we will show that the set of all f-subrings of A with the same identity has a smallest element and a greatest element with respect to the inclusion ordering. Also, we shall apply our main result to obtain a well known characterization of f-ring homomorphisms in terms of idempotent elements. PubDate: 2022-08-05 DOI: 10.1007/s00012-022-00792-3

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Abstract: Abstract The system of all congruence lattices of all algebras with fixed base set A forms a lattice with respect to inclusion, denoted by \(\mathcal {E}_A\) . Let A be finite. The meet-irreducible elements of \(\mathcal {E}_A\) are congruence lattices of monounary algebras. We assume that (A, f) has a connected subalgebra B such that B contains at least 3 cyclic elements and is meet-irreducible in \({\mathcal {E}}_B\) and we prove several sufficient conditions under which \({{\,\mathrm{Con}\,}}(A, f)\) is meet-irreducible in \({\mathcal {E}}_A\) . PubDate: 2022-08-05 DOI: 10.1007/s00012-022-00786-1

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Abstract: Abstract Ordered by set inclusion, the retracts of a lattice L together with the empty set form a bounded poset \(Ret (L)\) . By a grid we mean the direct product of two non-singleton finite chains. We prove that if G is a grid, then \(Ret (G)\) is a lattice. We determine the number of elements of \(Ret (G)\) . Some easy properties of retracts, retractions, and their kernels called retraction congruences of (mainly distributive) lattices are found. Also, we present several examples, including a 12-element modular lattice M such that \(Ret (M)\) is not a lattice. PubDate: 2022-08-02 DOI: 10.1007/s00012-022-00788-z

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Abstract: Abstract In this paper, we shed new light on the spectrum of the relation algebra we call \(A_{n}\) , which is obtained by splitting the non-flexible diversity atom of \(6_{7}\) into n symmetric atoms. Precisely, show that the minimum value in \(\text {Spec}(A_{n})\) is at most \(2n^{6 + o(1)}\) , which is the first polynomial bound and improves upon the previous bound due to Dodd and Hirsch (J Relat Methods Comput Sci 2:18–26, 2013). We also improve the lower bound to \(2n^{2} + 4n + 1\) , which is roughly double the trivial bound of \(n^{2} + 2n + 3\) . In the process, we obtain stronger results regarding \(\text {Spec}(A_{2}) =\text {Spec}(32_{65})\) . Namely, we show that 1024 is in the spectrum, and no number smaller than 26 is in the spectrum. Our improved lower bounds were obtained by employing a SAT solver, which suggests that such tools may be more generally useful in obtaining representation results. PubDate: 2022-07-30 DOI: 10.1007/s00012-022-00791-4

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Abstract: Abstract Câmpeanu and Ho (2004) determined the maximum finite state complexity of finite languages, building on work of Champarnaud and Pin (1989). They stated that it is very difficult to determine the number of maximum-complexity languages. Here we give a formula for this number. We also generalize their work from languages to functions on finite sets. PubDate: 2022-07-30 DOI: 10.1007/s00012-022-00785-2

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Abstract: Abstract 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, this approach allows us to represent congruences and elements of such algebras as the subsets of upward closed subsets of these posets with some special properties. PubDate: 2022-07-13 DOI: 10.1007/s00012-022-00787-0

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Abstract: Abstract 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 some \(r \in A\) and C is generated by B and r. For the class \({\mathcal {G}}\) of groups, we determine the mpe’s in \(S_{{\mathcal {G}}}(\{0\},{\mathbb {Q}})\) using invariants of Beaumont and Zuckerman and show that these (plus utilization of a Hamel basis) determine the mpe’s in \(S_{{\mathcal {G}}}(\{0\},{\mathbb {R}})\) . Finally, we show that the latter yield some (not all) of the minimum proper essential extensions in \(\mathbf {W}^{*}\) , the category of Archimedean \(\ell \) -groups with strong order unit and unit-preserving \(\ell \) -group homomorphisms. PubDate: 2022-07-07 DOI: 10.1007/s00012-022-00784-3

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Abstract: Abstract 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. PubDate: 2022-06-28 DOI: 10.1007/s00012-022-00776-3

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Abstract: Abstract 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 in a way what makes them conservative extensions of topological transitivity defined by G.D. Birkhoff. We describe the way the different properties are related to each other in pointfree topology. PubDate: 2022-06-28 DOI: 10.1007/s00012-022-00783-4

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Abstract: Abstract 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 functions. For complete representations, we show that a representation is meet complete if and only if it is join complete. Then we show that the class of completely representable algebras is precisely the class of atomic and representable algebras. As a corollary, the same properties axiomatise the class of algebras completely representable by injective partial functions. The universal-existential-universal axiomatisation this yields for these complete representation classes is the simplest possible, in the sense that no existential-universal-existential axiomatisation exists. PubDate: 2022-06-27 DOI: 10.1007/s00012-022-00775-4

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Abstract: Abstract It was proved by the authors that the quasivariety of quasi-Stone algebras \(\mathbf {Q}_{\mathbf {1,2}}\) is finite-to-finite universal relative to the quasivariety \(\mathbf {Q}_{\mathbf {2,1}}\) contained in \(\mathbf {Q}_{\mathbf {1,2}}\) . In this paper, we prove that \(\mathbf {Q}_{\mathbf {1,2}}\) is not Q-universal. This provides a positive answer to the following long standing open question: Is there a quasivariety that is relatively finite-to-finite universal but is not Q-universal' PubDate: 2022-06-27 DOI: 10.1007/s00012-022-00782-5

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Abstract: 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 of graphs by their endomorphism semigroups. PubDate: 2022-06-27 DOI: 10.1007/s00012-022-00780-7

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Abstract: Abstract In this paper we study prime spectra of commutative BCK-algebras. We give a new construction for commutative BCK-algebras using rooted trees, and determine both the ideal lattice and prime ideal lattice of such algebras. We prove that the spectrum of any commutative BCK-algebra is a locally compact generalized spectral space which is compact if and only if the algebra is finitely generated as an ideal. Further, we show that if a commutative BCK-algebra is involutory, then its spectrum is a Priestley space. Finally, we consider the functorial properties of the spectrum and define a functor from the category of commutative BCK-algebras to the category of distributive lattices with zero. We give a partial answer to the question: what distributive lattices lie in the image of this functor' PubDate: 2022-06-27 DOI: 10.1007/s00012-022-00779-0

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Abstract: Abstract 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. PubDate: 2022-06-27 DOI: 10.1007/s00012-022-00781-6

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Abstract: Abstract 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 totally ordered ring. PubDate: 2022-06-25 DOI: 10.1007/s00012-022-00778-1