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Title Expanding Belnap: dualities for a new class of default bilattices Author info Andrew P. K. Craig, Brian A. Davey, Miroslav Haviar Author Craig Andrew, P. K. (34%)
Co-authors Davey Brian A. (33%)
Haviar Miroslav 1965- (33%) UMBFP10 - Katedra matematiky
Source document Algebra universalis. Vol. 81, no. 4 (2020), pp. 1-26. - Basel : Springer Nature Switzerland AG, 2020 Keywords natural duality výsledky - results algebras Form. Descr. články - journal articles Language English Country Switzerland Annotation Bilattices provide an algebraic tool with which to model simultaneously knowledge and truth. They were introduced by Belnap in 1977 in a paper entitled How a computer should think. Belnap argued that instead of using a logic with two values, for ‘true’ and ‘false’, a computer should use a logic with two further values, for ‘contradiction’ and ‘no information´. The resulting structure is equipped with two lattice orders, a knowledge order and a truth order, and hence is called a bilattice. Prioritised default bilattices include not only values for ‘true’, ‘false’, ‘contradiction’ and ‘no information’, but also indexed families of default values for simultaneous modelling of degrees of knowledge and truth. We focus on a new family of prioritised default bilattices: Jn, for all natural numbers n. The bilattice J0 is precisely Belnap’s seminal example. We obtain a multisorted duality for the variety generated by Jn, and separately a single sorted duality for the quasivariety generated by Jn. The main tool for both dualities is a unified approach that enables us to identify the meet-irreducible elements of the appropriate subuniverse lattices. Our results provide an interesting example where the multi-sorted duality for the variety has a simpler structure than the single-sorted duality for the quasivariety URL Link na zdrojový dokument Public work category ADC No. of Archival Copy 48507 Catal.org. BB301 - Univerzitná knižnica Univerzity Mateja Bela v Banskej Bystrici Database xpca - PUBLIKAČNÁ ČINNOSŤ References PERIODIKÁ-Súborný záznam periodika 
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Title Bohr compactifications of algebras and structures Author info B. A. Davey, M. Haviar, H. A. Priestley Author Davey Brian A. (34%)
Co-authors Haviar Miroslav 1965- (33%) UMBFP10 - Katedra matematiky
Priestley Hilary A. (33%)
Source document Applied Categorical Structures. Vol. 25, no. 3 (2017), pp. 403-430. - Dordrecht : Springer, 2017 Person keywords Bohr Niels dánsky fyzik 1885-1962 Keywords natural duality natural extension distributive lattices Stone-Čechova kompaktifikácia - Stone-Čech compactification Language English Country Netherlands systematics 51 Annotation This paper provides a unifying framework for a range of categorical constructions characterised by universal mapping properties, within the realm of compactifications of discrete structures. Some classic examples fit within this broad picture: the Bohr compactification of an abelian group via Pontryagin duality, the zero-dimensional Bohr compactification of a semilattice, and the Nachbin order-compactification of an ordered set. The notion of a natural extension functor is extended to suitable categories of structures and such a functor is shown to yield a reflection into an associated category of topological structures. Our principal results address reconciliation of the natural extension with the Bohr compactification or its zero-dimensional variant. In certain cases the natural extension functor and a Bohr compactification functor are the same; in others the functors have different codomains but may agree on all objects. Coincidence in the stronger sense occurs in the zero-dimensional setting precisely when the domain is a category of structures whose associated topological prevariety is standard. It occurs, in the weaker sense only, for the class of ordered sets and, as we show, also for infinitely many classes of ordered structures. Coincidence results aid understanding of Bohr-type compactifications, which are defined abstractly. Ideas from natural duality theory lead to an explicit description of the natural extension which is particularly amenable for any prevariety of algebras with a finite, dualisable, generator. Examples of such classes-often varieties-are plentiful and varied, and in many cases the associated topological prevariety is standard. Public work category ADM No. of Archival Copy 39736 Database xpca - PUBLIKAČNÁ ČINNOSŤ References PERIODIKÁ-Súborný záznam periodika unrecognised
Title Reconciliation of approaches to the construction of canonical extensions of bounded lattices Par.title Zladenie prístupov ku konštrukcii kanonického rozšírenia ohraničeného zväzu Author info Andrew Craig, Miroslav Haviar Author Craig Andrew, P. K. (50%)
Co-authors Haviar Miroslav 1965- (50%) UMBFP10 - Katedra matematiky
Source document Mathematica Slovaca. Vol. 64, no. 3 (2014), pp. 1335-1356. - Bratislava : Slovenská akadémia vied, Matematický ústav SAV, 2014 Keywords kanonické rozšírenia topologická reprezentácia Galoisova väzba canonical extension natural duality Galois connection Language English Country Slovak Republic systematics 51 Public work category ADN No. of Archival Copy 31648 Repercussion category DÜNTSCH, Ivo - KWUIDA, Léonard - OROWSKA, Ewa. A discrete representation for dicomplemented lattices. In Fundamenta informaticae. ISSN 0169-2968, 2017, vol. 156, no. 3-4, pp. 281-295.
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Title A fresh perspective on canonical extensions for bounded lattices Par.title Nová perspektíva ohľadne kanonických rozšírení ohraničených zväzov Author info A. P. K. Craig, M. Haviar, H. A. Priestley Author Craig Andrew, P. K. (34%)
Co-authors Haviar Miroslav 1965- (33%) UMBFP10 - Katedra matematiky
Priestley Hilary A. (33%)
Source document Applied Categorical Structures. Vol. 21, no. 6 (2013), pp. 725-749. - Dordrecht : Springer, 2013 Keywords kanonické rozšírenia prirodzená dualita topologická reprezentácia canonical extension natural duality topological representation Language English Country Netherlands systematics 544.022 Annotation Kanonické rozšírenia algebier majú pôvod v klasických prácach B. Jónssona a A. Tarského (1951-52) o Booleových algebrách s operátormi. Z pohľadu logiky je význam kanonických rozšírení v tom, že pre mnohé logiky hrajú fundamentálnu úlohu vo vetách o úplnosti – kanonicita (znamenajúca, že algebraické identity sú zachované pri konštrukcii kanonických rozšírení) algebraických modelov logík korešponduje s úplnosťou logík. Prezentovaná je nová konštrukcia kanonických rozšírení ohraničených zväzov, ktorá je v duchu teórie prirodzených dualít. Na úrovni objektov je kanonické rozšírenie zväzu získané podobne ako v distributívnom prípade, kde sa používa Priestleyovej reprezentácia (1970). V nedistributívnom prípade je využitá topologická reprezentácia zväzov od Miroslava Ploščicu (1995), ktorá je prezentáciou klasickej Urquhartovej reprezentácie zväzov (1978) v duchu prirodzených dualít. Na úrovni morfizmov je využitá dualita Allweina a Hartonasa (1993) Public work category ADM No. of Archival Copy 27747 Repercussion category HARTONAS, Chrysafis. Order-dual relational semantics for non-distributive propositional logics : a general framework. In Journal of philosophical logic. ISSN 0022-3611, 2018, vol. 47, no. 1, pp. 67-94.
HARTONAS, Chrysafis. Order-dual relational semantics for non-distributive propositional logics. In Logic journal of the IGPL. ISSN 1367-0751, 2017, vol. 25, no. 2, pp. 145-182.
Catal.org. BB301 - Univerzitná knižnica Univerzity Mateja Bela v Banskej Bystrici Database xpca - PUBLIKAČNÁ ČINNOSŤ References PERIODIKÁ-Súborný záznam periodika unrecognised
Title Natural dualities in partnership Author info Brian A. Davey, Miroslav Haviar, Hilary A. Priestley Author Davey Brian A. (34%)
Co-authors Haviar Miroslav 1965- (33%) UMBFP10 - Katedra matematiky
Priestley Hilary A. (33%)
Source document Applied Categorical Structures. Vol. 20, no. 6 (2012), pp. 583-602. - Dordrecht : Springer, 2012 Keywords prirodzená dualita prirodzené rozšírenie kanonické rozšírenia natural duality natural extension canonical extension Language English Country Netherlands systematics 512 Public work category ADE No. of Archival Copy 23307 Catal.org. BB301 - Univerzitná knižnica Univerzity Mateja Bela v Banskej Bystrici Database xpca - PUBLIKAČNÁ ČINNOSŤ References PERIODIKÁ-Súborný záznam periodika article
Title Modified Priestley dualities as natural dualities Par.title Modifikované Priestlyovej duality ako prirodzené duality Author info Brian A. Davey, Miroslav Haviar Author Davey Brian A. (50%)
Co-authors Haviar Miroslav 1965- (50%) UMBFP10 - Katedra matematiky
Source document Lattice Theory: Foundation. S. 434-437. - Basel : Springer, 2011 / Grätzer George 1936- Keywords prirodzená dualita Priestleyovská dualita natural duality Priestley duality Language English Country Switzerland systematics 512 Public work category AEC No. of Archival Copy 19974 Catal.org. BB301 - Univerzitná knižnica Univerzity Mateja Bela v Banskej Bystrici Database xpca - PUBLIKAČNÁ ČINNOSŤ article
Title Transferral of entailment in duality theory: dualisability Author info Maria Joao Gouveia, Miroslav Haviar Author Gouveia Maria Joao (50%)
Co-authors Haviar Miroslav 1965- (50%) UMBFP10 - Katedra matematiky
Source document Czechoslovak mathematical journal. Vol. 61, no. 1 (2011), pp. 41-63. - Prague : Institute of mathematics, 2011 Keywords prirodzená dualita entailment dualizovateľnosť retrakcia natural duality dualisability retraction endodualisability Language English Country Germany systematics 512 Public work category ADE No. of Archival Copy 18310 Catal.org. BB301 - Univerzitná knižnica Univerzity Mateja Bela v Banskej Bystrici Database xpca - PUBLIKAČNÁ ČINNOSŤ References PERIODIKÁ-Súborný záznam periodika article
Title Multisorted dualisability: change of base Author info Brian A. Davey ... [et al.] Author Davey Brian A. (25%)
Co-authors Gouveia M. J. (25%)
Haviar Miroslav 1965- (25%) UMBFP10 - Katedra matematiky
Priestley Hilary A. (25%)
Source document Algebra Universalis. Vol. 66, no. 4 (2011), pp. 331-336. - Cham : Springer Nature Switzerland AG, 2011 Keywords prirodzená dualita dualizovateľnosť natural duality dualisability Language English Country Switzerland systematics 512 Annotation It is proved that if a quasivariety A generated by a finite family M of finite algebras has a multisorted duality based on M, then A has a multisorted duality based on any finite family of finite algebras that generates it Public work category ADE No. of Archival Copy 21511 Catal.org. BB301 - Univerzitná knižnica Univerzity Mateja Bela v Banskej Bystrici Database xpca - PUBLIKAČNÁ ČINNOSŤ References PERIODIKÁ-Súborný záznam periodika unrecognised
Title Natural extensions and profinite completions of algebras Author info B. A. Davey ... [et al.] Author Davey Brian A. (25%)
Co-authors Gouveia M. J. (25%)
Haviar Miroslav 1965- (25%) UMBFP10 - Katedra matematiky
Priestley Hilary A. (25%)
Source document Algebra Universalis. Vol. 66, no. 3 (2011), pp. 205-241. - Cham : Springer Nature Switzerland AG, 2011 Keywords prirodzené rozšírenie prirodzená dualita kanonické rozšírenia profinite completion natural extension natural duality canonical extension Language English Country Switzerland systematics 51 Annotation The paper investigates profinite completions of residually finite algebras, drawing on ideas from the theory of natural dualities. Given a class A = ISP(M), where M is a set, not necessarily finite, of finite algebras, it is shown that each algebra in the class A embeds as a topologically dense subalgebra of its natural extension, and that this natural extension is isomorphic, topologically and algebraically, to the profinite completion of the original algebra. In addition it is shown how the natural extension may be concretely described as a certain family of relation-preserving maps; in the special case that M is finite and the class A possesses a single-sorted or multisorted natural duality, the relations to be preserved can be taken to be those belonging to a dualising set. For an algebra belonging to a finitely generated variety of lattice-based algebras, it is known that the profinite completion coincides with the canonical extension. In this situation the natural extension provides a new concrete realisation of the canonical extension, generalising the well-known representation of the canonical extension of a bounded distributive lattice as the lattice of up-sets of the underlying ordered set of its Priestley dual. The paper concludes with a survey of classes of algebras to which the main theorems do, and do not, apply Public work category ADE No. of Archival Copy 20292 Repercussion category VOSMAER, Jacob. Logic, algebra and topology : investigations into canonical extensions, duality theory and point-free topology. Amsterdam : Institute for Logic, Language and Computation, 2010. 255 s. ISBN 978-90-5776-214-7.
Catal.org. BB301 - Univerzitná knižnica Univerzity Mateja Bela v Banskej Bystrici Database xpca - PUBLIKAČNÁ ČINNOSŤ References PERIODIKÁ-Súborný záznam periodika unrecognised
Title Transferral of entailment in duality theory II: strong dualisability Author info Maria Joao Gouveia, Miroslav Haviar Author Gouveia Maria Joao (50%)
Co-authors Haviar Miroslav 1965- (50%) UMBFP10 - Katedra matematiky
Source document Czechoslovak mathematical journal. Vol. 61, no. 2 (2011), s. 401-417. - Prague : Institute of mathematics, 2011 Keywords prirodzená dualita entailment dualizovateľnosť retrakcia natural duality dualisability retraction endodualisability Language English Country Germany systematics 512 Public work category ADE No. of Archival Copy 19015 Catal.org. BB301 - Univerzitná knižnica Univerzity Mateja Bela v Banskej Bystrici Database xpca - PUBLIKAČNÁ ČINNOSŤ References PERIODIKÁ-Súborný záznam periodika article