Papers by Miroslav Haviar
Vysokoškolská učebnica "Rozvoj kritického myslenia koučovacím prístupom vo vysokoškolskom prostre... more Vysokoškolská učebnica "Rozvoj kritického myslenia koučovacím prístupom vo vysokoškolskom prostredí", ktorej autormi sú Lenka Theodoulides a kol., je šírená pod licenciou Creative Commons Attribution 4.0 International Licence CC BY-NC. Lenka Theodoulides a kol. nekomerčné použitie .
Critical thinking as a key skill of every teacher is essential in guiding students through how to... more Critical thinking as a key skill of every teacher is essential in guiding students through how to analyse, understand and evaluate vast amounts of information, how to argue, solve problems, recognize new possibilities and form their own opinion. More precisely, this book aims to present several practical exercises, techniques, case studies and examples which can be used in all educational classes with activities including tutoring students, preparing a final theses at all three stages of university study, and also in research activities of students as well as teachers. Our approach, to use elements of coaching as a fundamental base to foster critical thinking, at the HEI is fundamentally based upon "questioning". The ability to develop a meaningful and reflective discussion between teacher and students consists of open-ended questions which are a key prerequisite to developing critical thinking skills. We would also like to acknowledge the fact that our inspiration and practical abilities of critical thinking and coaching were sharpened during our attendance of several training workshops provided by local as well as foreign entities, e.g. Živica Zaježova, PDCS Bratislava, SAKO (Slovak Coach Association), Solution Focus, Switzerland and University of Cambridge. Some of the activities have been developed and tested in cooperation with students from the
Norma
The aim of this theoretical study is to identify the 21st century skills in the curricular docume... more The aim of this theoretical study is to identify the 21st century skills in the curricular documents for lower secondary education in Slovakia, and to analyse the overlap between the 21st century skills and aims of national educational standards for lower secondary education (ISCED 2) in Slovakia. The main method of the study is document analysis. The analysis shows that the Slovak educational standards are primarily focused on the development of knowledge and cognitive skills. Significantly less represented is development of interpersonal and intrapersonal skills. The results show that there are overall not sufficiently many references to the 21st century skills within the educational standards. This yields a crucial problem, as majority of Slovak teachers still consider the educational standards to be the basic part of curriculum. In spite of certain natural limitations, the study provides new information on the insufficient representation of the 21st century skills in the educati...
Cubo (Temuco)
Dual digraphs of finite join-semidistributive lattices, meet-semidistributive lattices and semidi... more Dual digraphs of finite join-semidistributive lattices, meet-semidistributive lattices and semidistributive lattices are characterised. The vertices of the dual digraphs are maximal disjoint filter-ideal pairs of the lattice. The approach used here combines representations of arbitrary lattices due to Urquhart (1978) and Ploščica (1995). The duals of finite lattices are mainly viewed as TiRS digraphs as they were presented and studied in Craig--Gouveia--Haviar (2015 and 2022). When appropriate, Urquhart's two quasi-orders on the vertices of the dual digraph are also employed. Transitive vertices are introduced and their role in the domination theory of the digraphs is studied. In particular, finite lattices with the property that in their dual TiRS digraphs the transitive vertices form a dominating set (respectively, an in-dominating set) are characterised. A characterisation of both finite meet-and join-semidistributive lattices is provided via minimal closure systems on the se...
Categories and General Algebraic Structures with Applications
Bilattices, which provide an algebraic tool for simultaneously modelling knowledge and truth, wer... more Bilattices, which provide an algebraic tool for simultaneously modelling knowledge and truth, were introduced by N. D. Belnap in a 1977 paper entitled How a computer should think. Prioritised default bilattices include not only Belnap's four values, for 'true' (), 'false'(), 'contradiction' (⊤) and 'no information' (⊥), but also indexed families of default values for simultaneously modelling degrees of knowledge and truth. Prioritised default bilattices have applications in a number of areas including artificial intelligence.
Algebra universalis, 2022
Canonical extensions of lattice-based algebras originated in the 1951-52 papers of B. Jónsson and... more Canonical extensions of lattice-based algebras originated in the 1951-52 papers of B. Jónsson and A. Tarski on Boolean algebras with operators. When the members of a variety of lattice-based algebras are algebraic models of a logic, canonicity (meaning that algebraic identities are preserved when constructing canonical extensions of algebras) leads to completeness for the associated logic. The concept has been intensively studied for distributive lattice expansions, and more generally for lattice and even poset expansions, during the last 25 years (M. Gehrke, J. Harding, B. Jónsson, A. Palmigiano, H.A. Priestley, Y. Venema, and others). We present a brief overview of the studies of the canonical extensions of lattices.
Abstract. We consider properties of the graphs that arise as duals of bounded lattices in Plošč... more Abstract. We consider properties of the graphs that arise as duals of bounded lattices in Ploščica’s representation via maximal partial maps into the two-element set. We introduce TiRS graphs which abstract those duals of bounded lattices. We demonstrate their one-to-one correspon-dence with so–called TiRS frames which are a subclass of the class of RS frames introduced by Gehrke to represent perfect lattices. This yields a dual representation of finite lattices via finite TiRS frames, or equiv-alently finite TiRS graphs, which generalises the well-known Birkhoff dual representation of finite distributive lattices via finite posets. By us-ing both Ploščica’s and Gehrke’s representations in tandem we present a new construction of the canonical extension of a bounded lattice. We present two open problems that can be of interest to researchers working in this area.
ABSTRACT. We provide new insights into the relationship between different constructions of the ca... more ABSTRACT. We provide new insights into the relationship between different constructions of the canonical extension of a bounded lattice. This follows on from the recent construction of the canonical extension using Ploščica’s maximal partial maps into the two-element set by Craig, Haviar and Priestley (2012). We show how this complete lattice of maps is isomorphic to the stable sets of Urquhart’s representation and to the concept lattice of a specific context, and how to translate our construction to the original construction of Gehrke and Harding (2001). In addition, we identify the completely join- and completely meet-irreducible elements of the complete lattice of maximal partial maps. c©2014
Algebra universalis, 2020
Bilattices provide an algebraic tool with which to model simultaneously knowledge and truth. They... more 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' (t) and 'false' (f), 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' (t0), 'false' (f0), 'contradiction' and 'no information', but also indexed families of default values, t1,. .. , tn and f1,. .. , fn, for simultaneous modelling of degrees of knowledge and truth. We focus on a new family of prioritised default bilattices: Jn, for n ∈ ω. The bilattice J0 is precisely Belnap's seminal example. We address mathematical rather than logical aspects of our prioritised default bilattices. We obtain a single-sorted topological representation for the bilattices in the quasivariety J n generated by Jn, and separately a multi-sorted topological representation for the bilattices in the variety Vn generated by Jn. 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.
Applied Categorical Structures, 2016
This paper provides a unifying framework for a range of categorical constructions characterised b... more 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.
Studia Logica, 2004
This paper illustrates how Priestley duality can be used in the transfer of an optimal natural du... more This paper illustrates how Priestley duality can be used in the transfer of an optimal natural duality from a minimal generating algebra for a quasi-variety to other generating algebras. Detailed calculations are given for the quasi-variety ISP(4) of Kleene algebras and the quasi-varieties Bn of pseudocomplemented distributive lattices (n 1).
Applied Categorical Structures, 2011
Traditionally in natural duality theory the algebras carry no topology and the objects on the dua... more Traditionally in natural duality theory the algebras carry no topology and the objects on the dual side are structured Boolean spaces. Given a duality, one may ask when the topology can be swapped to the other side to yield a partner duality (or, better, a dual equivalence) between a category of topological algebras and a category of structures. A prototype for this procedure is provided by the passage from Priestley duality for bounded distributive lattices to Banaschewski duality for ordered sets. Moreover, the partnership between these two dualities yields as a spinoff a factorisation of the functor sending a bounded distributive lattice to its natural extension, alias, in this case, the canonical extension or profinite completion. The main theorem of this paper validates topology swapping as a uniform way to create new dual adjunctions and dual equivalences: we prove that, for every finite algebra of finite type, each dualising alter ego gives rise to a partner duality. We illustrate the theorem via a variety of natural dualities, some classic and some less familiar.
We show that, in many cases, if D and M are finite algebras which generate the same quasi-variety... more We show that, in many cases, if D and M are finite algebras which generate the same quasi-variety ,, then a strong duality for , based on D may be transferred to a strong duality for , based on M by the addition some endomorphisms of M and just one further partial operation. This additional operation exhibits the schizophrenia so typical of the theory of natural dualities. We show how the result may be applied to yield an efficient strong duality in the case when M is a distributive lattice, a semilattice or an abelian group.
Algebra universalis, 2011
We prove that if a quasivariety A generated by a finite family M of finite algebras has a multiso... more We prove 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.
Algebra universalis, 2011
This paper investigates profinite completions of residually finite algebras, drawing on ideas fro... more This 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 A ∈ A embeds as a topologically dense subalgebra of a topological algebra n A (A) (its natural extension), and that n A (A) is isomorphic, topologically and algebraically, to the profinite completion of A. 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 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 upsets 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.
Algebra universalis, 2008
The finite endodualisable double Stone algebras are characterised, and every finite endoprimal do... more The finite endodualisable double Stone algebras are characterised, and every finite endoprimal double Stone algebra is shown to be endodualisable.
Algebra Universalis, 1995
L. M/trki and R. P6schel have characterised the endoprimal distributive lattices as those which a... more L. M/trki and R. P6schel have characterised the endoprimal distributive lattices as those which are not relatively complemented. The theory of natural dualities implies that any finite algebra A on which the endomorphisms of A yield a duality on the quasivariety DSP(A) is necessarily endoprimal. This note investigates endodualisability for finite distributive lattices, and shows, in a manner which elucidates Mfirki and P6schel's proof, that it is equivalent to endoprimality. Let (A; F) be an algebra and denote its endomorphism monoid by End A. Then A is said to be endoprimal if every finitary function on A commuting with each element of End A is a term function, and k-endoprimal (k >_ 1) if every function of arity not greater than k commuting with all endomorphisms is a term function. In [15], L. Mfirki and R. P6schel proved that the following statements are equivalent for a non-trivial distributive lattice (L; v, A): (1) L is endoprimal, and (2) L is not relatively complemented. The contrapositive of the implication (1) ~ (2) is easy (by exploiting relative complementation, as a ternary function). As Mfirki and P6schel point out, their result shows that an endoprimal algebra need not generate a quasivariety with the kind of nice structural properties that are obtained when primality is extended in other ways, notably to quasiprimality (see [16]). Thus the occurrence and role of endoprimal algebras seems a little mysterious, and examples are rather scarce. However there is one situation in which endoprimal algebras arise very naturally, through duality theory. When we refer to a duality, we shall always mean a natural duality in the sense of Davey and Werner. For our purposes, the recent survey [4] is the most convenient reference. A finite algebra A is said to be endodualisable if End A yields a duality on d = I~P(A), in the sense defined in [4], Section 2. If this
arXiv: Category Theory, 2019
This is a survey on selected developments in the theory of natural dualities where the author had... more This is a survey on selected developments in the theory of natural dualities where the author had the opportunity to make with his foreign colleagues several breakthroughs and move the theory forward. It is aimed as author's reflection on his works on the natural dualities in Oxford and Melbourne over the period of twenty years 1993-2012 (before his attention with the colleagues in universal algebra and lattice theory has been fully focused on the theory of canonical extensions and the theory of bilattices). It is also meant as a remainder that the main problems of the theory of natural dualities, Dualisability Problem and Decidability Problem for Dualisability, remain still open. Theory of natural dualities is a general theory for quasi-varieties of algebras that generalizes `classical' dualities such as Stone duality for Boolean algebras, Pontryagin duality for abelian groups, Priestley duality for distributive lattices, and Hofmann-Mislove-Stralka duality for semilattices...
Applied Categorical Structures, 2012
Your article is protected by copyright and all rights are held exclusively by Springer Science+Bu... more Your article is protected by copyright and all rights are held exclusively by Springer Science+Business Media B.V.. This e-offprint is for personal use only and shall not be selfarchived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com".
Mathematica Slovaca, 2014
We provide new insights into the relationship between different constructions of the canonical ex... more We provide new insights into the relationship between different constructions of the canonical extension of a bounded lattice. This follows on from the recent construction of the canonical extension using Ploščica’s maximal partial maps into the two-element set by Craig, Haviar and Priestley (2012). We show how this complete lattice of maps is isomorphic to the stable sets of Urquhart’s representation and to the concept lattice of a specific context, and how to translate our construction to the original construction of Gehrke and Harding (2001). In addition, we identify the completely join- and completely meet-irreducible elements of the complete lattice of maximal partial maps.
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Papers by Miroslav Haviar