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Title Canonical extensions of lattices are more than perfect Author info Andrew P. K. Craig, Maria J. Gouveia, Miroslav Haviar Author Craig Andrew, P. K. (34%)
Co-authors Gouveia Maria Joao (33%)
Haviar Miroslav 1965- (33%) UMBFP10 - Katedra matematiky
Source document Algebra Universalis. Vol. 83, no. 2 (2022), pp. [1-17]. - Basel : Springer Nature Switzerland AG, 2022 Keywords kanonické rozšírenia matematika - mathematics Form. Descr. články - journal articles Language English Country Switzerland Annotation In a paper published in 2015, we introduced TiRS graphs and TiRS frames to create a new natural setting for duals of canonical exten sions 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. URL Link na plný text Public work category ADC No. of Archival Copy 51538 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 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.
Catal.org. BB301 - Univerzitná knižnica Univerzity Mateja Bela v Banskej Bystrici Database xpca - PUBLIKAČNÁ ČINNOSŤ References PERIODIKÁ-Súborný záznam periodika article
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 
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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 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 Boolean topological distributive lattices and canonical extensions Author info B. A. Davey, Miroslav Haviar, H. A. Priestley Author Davey Brian A. (34%)
Co-authors Haviar Miroslav 1965- (33%) UMBUV01 - Ústav vedy a výskumu
Priestley Hilary A. (33%)
Source document Applied Categorical Structures. Vol. 15, no. 3 (2007), pp. 225-241. - Dordrecht : Springer, 2007 Keywords topologický zväz Priestleyovská dualita kanonické rozšírenia prokonečné rozšírenie topological lattice Priestley duality canonical extension profinite completion Language English Country Netherlands systematics 515.1 Public work category ADE No. of Archival Copy 6651 Repercussion category JOHANSEN, Sarah M. Natural dualities for three classes of relational structures. In Algebra Universalis. ISSN 0002-5240, 2010, vol. 63, no. 2-3, pp. 149-170.
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.
RICE, Brian. Intervals of the Muchnik lattice. In Fundamenta mathematicae. ISSN 0016-2736, 2018, vol. 241, no. 2, pp. 109-126.
Catal.org. BB301 - Univerzitná knižnica Univerzity Mateja Bela v Banskej Bystrici Database xpca - PUBLIKAČNÁ ČINNOSŤ References PERIODIKÁ-Súborný záznam periodika article
Title Canonical extensions of stone and double Stone algebras: the natural way Author info Miroslav Haviar, H. A. Priestley Author Haviar Miroslav 1965- (50%) UMBUV01 - Ústav vedy a výskumu
Co-authors Priestley Hilary A. (50%)
Source document Mathematica Slovaca. Vol. 56, no. 1 (2006), pp. 53-78. - Bratislava : Slovenská akadémia vied, Matematický ústav SAV, 2006 Keywords Stoneova algebra kanonické rozšírenia teória dualít Stone algebra duality theory Language English Country Slovak Republic systematics 512 Public work category ADF No. of Archival Copy 4518 Repercussion category DÜNTSCH, Ivo - ORŁOWSKA, Ewa. Discrete dualities for double Stone algebras. In Studia logica. ISSN 0039-3215, 2011, vol. 99, no. 1, pp. 127-142.
DAVEY, B. A. Natural dualities for structures. In Acta Universitatis Matthiae Belii : series mathematics. No. 13. Banská Bystrica : Univerzita Mateja Bela, 2006. ISBN 80-8083-379-6, pp. 3-28.
Catal.org. BB301 - Univerzitná knižnica Univerzity Mateja Bela v Banskej Bystrici Database xpca - PUBLIKAČNÁ ČINNOSŤ References PERIODIKÁ-Súborný záznam periodika article