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On k-ary parts of maximal clones Algebra Univers. (IF 0.6) Pub Date : 2024-03-18
Abstract The main problem of clone theory is to describe the clone lattice for a given basic set. For a two-element basic set this was resolved by E.L. Post, but for at least three-element basic set the full structure of the lattice is still unknown, and the complete description in general is considered to be hopeless. Therefore, it is studied by its substructures and its approximations. One of the
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Submaximal clones over a three-element set up to minor-equivalence Algebra Univers. (IF 0.6) Pub Date : 2024-03-18 Albert Vucaj, Dmitriy Zhuk
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Generalized quasiorders and the Galois connection $${\textbf {End}}$$ – $$\varvec{{{\,\textrm{gQuord}\,}}}$$ Algebra Univers. (IF 0.6) Pub Date : 2024-03-18 Danica Jakubíková-Studenovská, Reinhard Pöschel, Sándor Radeleczki
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Some further results on pointfree convex geometry Algebra Univers. (IF 0.6) Pub Date : 2024-03-06 Changchun Xia
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Factor principal congruences and Boolean products in filtral varieties Algebra Univers. (IF 0.6) Pub Date : 2024-03-05 Brian A. Davey, Miroslav Haviar
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Joins and meets in effect algebras Algebra Univers. (IF 0.6) Pub Date : 2024-03-01 Grzegorz Bińczak, Joanna Kaleta, Andrzej Zembrzuski
It is known that every effect algebra can be represented as the effect algebra of perspectivity classes of some E-test space. We describe when there exists join and meet of two perspectivity classes of events of some algebraic E-test space. Moreover we give the formula for join and meet of perspectivity classes mentioned above, using only tests. We obtain an example of finite, non-homogeneous effect
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Hull classes in compact regular frames Algebra Univers. (IF 0.6) Pub Date : 2024-03-01 Papiya Bhattacharjee, Ricardo E. Carrera
\(\mathfrak {KReg}\) is the category of compact regular frames and frame homomorphisms. A class of \(\mathfrak {KReg}\) frames \(\textbf{H}\) is a hull class provided that: (i) \(\textbf{H}\) is closed under isomorphic copies; (ii) for every \(F \in \mathfrak {KReg}\) there exist an \(hF \in \textbf{H}\) and a morphism \(h_F\) such that \(F \overset{h_F}{\le }\ hF\) is essential; (iii) if \(F \overset{\phi
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Solution to two problems on ideally conjunctive join-semilattices Algebra Univers. (IF 0.6) Pub Date : 2024-02-22 Ao Shen, Qingguo Li
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On the Boolean algebra induced by a unital $$\ell $$ -group Algebra Univers. (IF 0.6) Pub Date : 2024-02-22 Soudabeh Karamdoust, Hassan Myrnouri, Mahmood Pourgholamhossein
The main goal of this paper is the study of the Boolean algebra of all characteristic elements in a unital \(\ell \)-group and we investigate some topological properties of it in the case that the unital \(\ell \)-group equipped with a link or positive filter topology. We also introduce the concept of Boolean region as a subset of a unital \(\ell \)-group.
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Was Ulam right? II: small width and general ideals Algebra Univers. (IF 0.6) Pub Date : 2024-02-17 Tanmay Inamdar, Assaf Rinot
We continue our study of Sierpiński-type colourings. In contrast to the prequel paper, we focus here on colourings for ideals stratified by their completeness degree. In particular, improving upon Ulam’s theorem and its extension by Hajnal, it is proved that if \(\kappa \) is a regular uncountable cardinal that is not weakly compact in L, then there is a universal witness for non-weak-saturation of
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Not all nilpotent monoids are finitely related Algebra Univers. (IF 0.6) Pub Date : 2024-02-14 Markus Steindl
A finite semigroup is finitely related (has finite degree) if its term functions are determined by a finite set of finitary relations. For example, it is known that all nilpotent semigroups are finitely related. A nilpotent monoid is a nilpotent semigroup with adjoined identity. We show that every 4-nilpotent monoid is finitely related. We also give an example of a 5-nilpotent monoid that is not finitely
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Bubble lattices I: Structure Algebra Univers. (IF 0.6) Pub Date : 2024-02-07
Abstract C. Greene introduced the shuffle lattice as an idealized model for DNA mutation and discovered remarkable combinatorial and enumerative properties of this structure. We attempt an explanation of these properties from a lattice-theoretic point of view. To that end, we introduce and study an order extension of the shuffle lattice, the bubble lattice. We characterize the bubble lattice both locally
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Finitely based congruence varieties Algebra Univers. (IF 0.6) Pub Date : 2024-01-12 Ralph Freese, Paolo Lipparini
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Semilinear De Morgan monoids and epimorphisms Algebra Univers. (IF 0.6) Pub Date : 2024-01-09 J. J. Wannenburg, J. G. Raftery
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Priestley-style duality for DN-algebras Algebra Univers. (IF 0.6) Pub Date : 2024-01-03 Luciano J. González
The aim of this article is to develop a Priestley-style duality for the variety of DN-algebras. In order to achieve this, we use the concept of free distributive lattice extension of a DN-algebra. We establish a connection with the Priestley duality for distributive lattices. Finally, we present topological descriptions for the lattice of filters, for the lattice of congruences, and for certain kinds
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The “zero-two” law in Orlicz–Kantorovich spaces Algebra Univers. (IF 0.6) Pub Date : 2024-01-03 Inomjon Ganiev, Farrukh Mukhamedov
Abstract The present paper deals with \(L_0\) -valued measures and the associated Orlicz–Kantorovich lattice. The considered Orlicz–Kantorovich lattice, endowed with the Luxemburg norm, is represented as a measurable bundle of classical Orlicz spaces associated with scalar measures. This kind of representation allows us to investigate positive contractions and apply the corresponding zero-two laws
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On a statement of O. Frink about free pseudo-complemented meet-semilattices Algebra Univers. (IF 0.6) Pub Date : 2023-12-14 Tibor Katriňák, Jaroslav Guričan
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Esakia duals of regular Heyting algebras Algebra Univers. (IF 0.6) Pub Date : 2023-11-21 Gianluca Grilletti, Davide Emilio Quadrellaro
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Majority-closed minions of Boolean functions Algebra Univers. (IF 0.6) Pub Date : 2023-11-22 Erkko Lehtonen
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Surjective polymorphisms of directed reflexive cycles Algebra Univers. (IF 0.6) Pub Date : 2023-11-17 Isabelle Larivière, Benoît Larose, David E. Pazmiño Pullas
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Choice-free topological duality for implicative lattices and Heyting algebras Algebra Univers. (IF 0.6) Pub Date : 2023-11-14 Chrysafis Hartonas
We develop a common semantic framework for the interpretation both of \({\textbf {IPC}}\), the intuitionistic propositional calculus, and of logics weaker than \({\textbf {IPC}}\) (substructural and subintuitionistic logics). To this end, we prove a choice-free representation and duality theorem for implicative lattices, which may or may not be distributive. The duality specializes to a choice-free
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Vaughan-Lee’s nilpotent loop of size 12 is finitely based Algebra Univers. (IF 0.6) Pub Date : 2023-11-14 Peter Mayr
From work of Vaughan-Lee in [12] it follows that if a finite nilpotent loop splits into a direct product of factors of prime power order, then its equational theory has a finite basis. Whether the condition on the direct decomposition is necessary has remained open since. In the same paper, Vaughan-Lee gives an explicit example of a nilpotent loop of order 12 that does not factor into loops of prime
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Efficient realizations of closure systems Algebra Univers. (IF 0.6) Pub Date : 2023-11-11 Robert E. Jamison
As is well-known, the subalgebras of any universal algebra form an algebraic closure system. Conversely, every algebraic closure system arises as the family of subalgebras of some universal algebra, but this algebra is far from uniquely determined. This paper investigates the realization of algebraic closure systems by algebras given either by a single operation or by operations of the lowest arity
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On automorphisms of categories with applications to universal algebraic geometry Algebra Univers. (IF 0.6) Pub Date : 2023-10-31 Grigori Zhitomirski
Let \({\mathcal {V}}\) be a variety of algebras of some type \(\Omega \). An interest to describing automorphisms of the category \(\Theta ^0 ({\mathcal {V}})\) of finitely generated free \({\mathcal {V}}\)-algebras was inspired by development of universal algebraic geometry founded by B. Plotkin. There are a lot of results on this subject. A common method of getting such results was suggested and
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A frame-theoretic perspective on Esakia duality Algebra Univers. (IF 0.6) Pub Date : 2023-09-30 G. Bezhanishvili, L. Carai, P. J. Morandi
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On the lattice of conatural classes of linear modular lattices Algebra Univers. (IF 0.6) Pub Date : 2023-09-21 Sebastián Pardo-Guerra, Hugo A. Rincón-Mejía, Manuel G. Zorrilla-Noriega, Francisco González-Bayona
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On the variety generated by generalized subreducts of Tarski’s algebras of relations Algebra Univers. (IF 0.6) Pub Date : 2023-09-01 Dmitry A. Bredikhin
In the paper, a basis of identities for the variety generated by the class of groupoids that are generalized subreducts of Tarski’s algebra of relations is found. It is also proved that the corresponding class of groupoids does not form a variety.
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Automorphisms and strongly invariant relations Algebra Univers. (IF 0.6) Pub Date : 2023-08-09 Ferdinand Börner, Martin Goldstern, Saharon Shelah
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The modal logic of abelian groups Algebra Univers. (IF 0.6) Pub Date : 2023-07-26 Sören Berger, Alexander Christensen Block, Benedikt Löwe
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Existential relations on infinite structures Algebra Univers. (IF 0.6) Pub Date : 2023-07-19 Boris A. Romov
We establish a criterion for a structure M on an infinite domain to have the Galois closure \({{\,\textrm{InvAut}\,}}(M)\) (the set all relations on the domain of M that are invariant to all automorphisms of M) defined via infinite Boolean combinations of infinite (constructed by infinite conjunction) existential relations from M. Based on this approach, we present criteria for quantifier elimination
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Topological representations of Lawson compact algebraic L-domains and Scott domains Algebra Univers. (IF 0.6) Pub Date : 2023-07-17 Longchun Wang, Xiangnan Zhou, Qingguo Li
In this paper, the relationships between two important subclasses of algebraic dcpos and topological spaces which may not be \(\textrm{T}_0\) are discussed. The concepts of CFF-spaces and strong CFF-spaces are introduced by considering the properties of their topological bases. With these concepts, Lawson compact algebraic L-domains and Scott domains are successfully represented in purely topological
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Weakly Schreier extensions for general algebras Algebra Univers. (IF 0.6) Pub Date : 2023-07-14 Graham Manuell, Nelson Martins-Ferreira
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Two results on Fremlin’s Archimedean Riesz space tensor product Algebra Univers. (IF 0.6) Pub Date : 2023-07-14 Gerard Buskes, Page Thorn
In this paper, we characterize when, for any infinite cardinal \(\alpha \), the Fremlin tensor product of two Archimedean Riesz spaces (see Fremlin in Am J Math 94:777–798, 1972) is Dedekind \(\alpha \)-complete. We also provide an example of an ideal I in an Archimedean Riesz space E such that the Fremlin tensor product of I with itself is not an ideal in the Fremlin tensor product of E with itself
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Strongly minimal Steiner systems II: coordinatization and quasigroups Algebra Univers. (IF 0.6) Pub Date : 2023-05-02 John T. Baldwin
Each strongly minimal Steiner k-system (M, R) (where is R is a ternary collinearity relation) can be ‘coordinatized’ in the sense of (Ganter–Werner 1975) by a quasigroup if k is a prime-power. We show this coordinatization is never definable in (M, R) and the strongly minimal Steiner k-systems constructed in (Baldwin–Paolini 2020) never interpret a quasigroup. Nevertheless, by refining the construction
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Admissible subsets and completions of ordered algebras Algebra Univers. (IF 0.6) Pub Date : 2023-04-28 Valdis Laan, Jianjun Feng, Xia Zhang
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Stone space partitions indexed by a poset Algebra Univers. (IF 0.6) Pub Date : 2023-04-27 Andrew B. Apps
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Topology of closure systems in algebraic lattices Algebra Univers. (IF 0.6) Pub Date : 2023-04-25 Niels Schwartz
Algebraic lattices are spectral spaces for the coarse lower topology. Closure systems in algebraic lattices are studied as subspaces. Connections between order theoretic properties of a closure system and topological properties of the subspace are explored. A closure system is algebraic if and only if it is a patch closed subset of the ambient algebraic lattice. Every subset X in an algebraic lattice
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Extremality, left-modularity and semidistributivity Algebra Univers. (IF 0.6) Pub Date : 2023-04-17 Henri Mühle
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Topological representation of double Boolean algebras Algebra Univers. (IF 0.6) Pub Date : 2023-03-20 Prosenjit Howlader, Mohua Banerjee
The article proves topological representations for some classes of double Boolean algebras (dBas). In particular, representation theorems characterising fully contextual and pure dBas are obtained. Duality results for fully contextual and pure dBas are also established.
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Quasi-Engel varieties of lattice-ordered groups Algebra Univers. (IF 0.6) Pub Date : 2023-03-20 Michael R. Darnel
We show that any ordered group satisfying the identity \([x_1^{k_1}, \ldots , x_n^{k_n}] = e\) must be weakly abelian and that when \(x_i \not = x_1\) for \(2 \le i \le n\), \(\ell \)-groups satisfying the identity \([x_1^n, \ldots , x_k^n] = e\) also satisfy the identity \((x \vee e)^{y^n} \le (x \vee e)^2\). These results are used to study the structure of \(\ell \)-groups satisfying identities of
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Congruence-simple multiplicatively idempotent semirings Algebra Univers. (IF 0.6) Pub Date : 2023-03-13 Tomáš Kepka, Miroslav Korbelář, Günter Landsmann
Let S be a multiplicatively idempotent congruence-simple semiring. We show that \(|S|=2\) if S has a multiplicatively absorbing element. We also prove that if S is finite then either \(|S|=2\) or \(S\cong {{\,\textrm{End}\,}}(L)\) or \(S^{op}\cong {{\,\textrm{End}\,}}(L)\) where L is the 2-element semilattice. It seems to be an open question, whether S can be infinite at all.
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Higher commutators in semigroups with zero Algebra Univers. (IF 0.6) Pub Date : 2023-03-07 Jelena Radović, Nebojša Mudrinski
We introduce the notion of the higher commutator of ideals in semigroups. For semigroups with zero, it is shown that the higher order commutator of Rees congruences is equal to the Rees congruence of the commutator of the corresponding ideals. We obtain that, for Rees congruences, higher order commutator is a composition of binary commutators. As a consequence, we prove that in semigroups with zero
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On the representation of ordered semigroups by transformations of ordered sets Algebra Univers. (IF 0.6) Pub Date : 2023-03-02 Michael Tsingelis
A transformation of an ordered set M is an isotone mapping of M into M. By a representation of an ordered semigroup S by transformations of an ordered set M we mean a homomorphism of S into the set of transformations of M, i.e. (since the set of transformations of M is an ordered semigroup) an isotone mapping from S into the set of transformations of M preserving the operations. We prove that this
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Catoids and modal convolution algebras Algebra Univers. (IF 0.6) Pub Date : 2023-02-25 Uli Fahrenberg, Christian Johansen, Georg Struth, Krzysztof Ziemiański
We show how modal quantales arise as convolution algebras \(Q^X\) of functions from catoids X, multisemigroups equipped with source and target maps, into modal quantales value or weight quantales Q. In the tradition of boolean algebras with operators we study modal correspondences between algebraic laws in X, Q and \(Q^X\). The catoids introduced generalise Schweizer and Sklar’s function systems and
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Varieties of ordered algebras as categories Algebra Univers. (IF 0.6) Pub Date : 2023-02-24 Jiří Adámek, Jiří Rosický
A variety is a category of ordered (finitary) algebras presented by inequations between terms. We characterize categories enriched over the category of posets which are equivalent to a variety. This is quite analogous to Lawvere’s classical characterization of varieties of ordinary algebras. We also study the relationship of varieties to discrete Lawvere theories, and varieties as concrete categories
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Adjunction of a strong unit to a hyper-archimedean lattice-ordered group Algebra Univers. (IF 0.6) Pub Date : 2023-02-13 Philip Scowcroft
This paper studies conditions in which a hyperarchimedean lattice-ordered group embeds into a hyperarchimedean lattice-ordered group with strong unit. While Conrad and Martinez showed that some hyperarchimedean lattice-ordered groups do not admit such embeddings, Section 3 presents a sufficient condition, in terms of the generalized Boolean algebra of principal \(\ell \)-ideals, for the existence of
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Lacunary series, algebraic normal forms, convolutions Algebra Univers. (IF 0.6) Pub Date : 2023-02-11 Arthur Knoebel
Conditions are found on an operation on a finite set for its transform to be lacunary, that is, missing many expected terms. Often the condition is that the operation preserves a relation. General operations are split into odd and even lacunary parts, or more generally, into several lacunary parts given by a non-binary parity. This is applied to classical Fourier transforms as well as algebraic normal
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What makes a Stone topological algebra Profinite Algebra Univers. (IF 0.6) Pub Date : 2023-01-29 Jorge Almeida, Herman Goulet-Ouellet, Ondřej Klíma
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Duality for normal lattice expansions and sorted residuated frames with relations Algebra Univers. (IF 0.6) Pub Date : 2023-01-29 Chrysafis Hartonas
We revisit the problem of Stone duality for lattices with quasioperators, presenting a fresh duality result. The new result is an improvement over that of our previous work in two important respects. First, the axiomatization of frames is now simplified, partly by incorporating Gehrke’s proposal of section stability for relations. Second, morphisms are redefined so as to preserve Galois stable (and
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Connected topological lattice-ordered groups Algebra Univers. (IF 0.6) Pub Date : 2023-01-10 Francis Jordan
We answer two open problems about lattice-ordered groups that admit a connected lattice-ordered group topology. We show that, in the general case, admitting a connected lattice-ordered group topology does not effect the algebraic structure of the lattice-ordered group. For example, admitting a connected lattice-ordered group topology does not imply that the lattice-ordered group is Archimedean or even
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Algebraic frames in which dense elements are above dense compact elements Algebra Univers. (IF 0.6) Pub Date : 2022-12-03 Themba Dube, Siphamandla Blose
A ring is called a zip ring (Carl Faith coined this term) if every faithful ideal contains a finitely generated faithful ideal. By first proving that a reduced ring is a zip ring if and only if every dense element of the frame of its radical ideals is above a compact dense element, we study algebraic frames with the property stated in the title. We call them zipped. They generalize the coherent frames
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Projectivity in (bounded) commutative integral residuated lattices Algebra Univers. (IF 0.6) Pub Date : 2022-11-29 Paolo Aglianò, Sara Ugolini
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On the number of universal algebraic geometries Algebra Univers. (IF 0.6) Pub Date : 2022-11-28 Erhard Aichinger, Bernardo Rossi
The algebraic geometry of a universal algebra \({\textbf{A}}\) is defined as the collection of solution sets of systems of term equations. Two algebras \({\textbf{A}}_1\) and \({\textbf{A}}_2\) are called algebraically equivalent if they have the same algebraic geometry. We prove that on a finite set A with \(|A|\) there are countably many algebraically inequivalent Mal’cev algebras and that on a finite
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Another look on tense and related operators Algebra Univers. (IF 0.6) Pub Date : 2022-11-01 Michal Botur, Jan Paseka, Richard Smolka
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The assembly of a pointfree bispace and its two variations Algebra Univers. (IF 0.6) Pub Date : 2022-10-08 Anna Laura Suarez
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
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Uniform continuity of pointfree real functions via farness and related Galois connections Algebra Univers. (IF 0.6) Pub Date : 2022-10-03 Ana Belén Avilez, Jorge Picado
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
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A theorem of Mumford and Ramanujam for universal algebras Algebra Univers. (IF 0.6) Pub Date : 2022-08-07 A. Clay, R. Padmanabhan
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Choice-free duality for orthocomplemented lattices by means of spectral spaces Algebra Univers. (IF 0.6) Pub Date : 2022-08-07 Joseph McDonald, Kentarô Yamamoto
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Monounary algebras containing subalgebras with meet-irreducible congruence lattice Algebra Univers. (IF 0.6) Pub Date : 2022-08-05 Lucia Janičková