arXiv subjects

Miroslav Haviar

Publications and source records attributed to Miroslav Haviar.

9 recordsLinked to original sources

arXiv2026

Dual spaces of lattices and semidistributive lattices

Birkhoff's 1937 dual representation of finite distributive lattices via finite posets was in 1970 extended to a dual representation of arbitrary distributive lattices via compact totally order-disconnected topological spaces by Priestley. This result enabled the development of natural duality theory in the 1980s by Davey and Werner, later on also in collaboration with Clark and Priestley. In 1978 Urquhart extended Priestley's representation to general lattices via compact doubly quasi-ordered topological spaces (L-spaces). In 1995 Ploščica presented Urquhart's representation in the spirit of natural duality theory by replacing, on the dual side, Urquhart's two quasiorders with a digraph relation generalising Priestley's order relation. In this paper we translate, following the spirit of natural duality theory, Urquhart's L-spaces into newly introduced \emph{Ploščica spaces}. We then prove that every Ploščica space is the dual space of some general lattice. Based on the authors' 2022 characterisation of finite join and meet semidistributive lattices via their dual digraphs, we initiate a study of general (possibly infinite) join and meet semidistributive lattices via their dual digraphs. We illustrate our results on examples and formulate three open problems.

math.RA↗

arXiv2026

Dual Ploščica spaces of ortholattices

We describe digraphs with topology which give dual representations of ortholattices. This is done via so-called dual Ploščica spaces of lattices. First, we improve the definition of Ploščica spaces from an earlier paper to give a straight and natural generalisation of the total order disconnectedness of Priestley spaces. Then we define the dual space of a general ortholattice as the dual Ploščica space of the lattice-reduct of the ortholattice equipped with a map representing the orthocomplement operation. We introduce an abstract ortho-Ploščica space capturing the properties of the dual space of an ortholattice, and we present dual representation theorems between general ortholattices and the ortho-Ploščica spaces. We illustrate our dual representations by examples.

math.RA↗

arXiv2023

Dual digraphs of finite meet-distributive and modular lattices

We describe the digraphs that are dual representations of finite lattices satisfying conditions related to meet-distributivity and modularity. This is done using the dual digraph representation of finite lattices by Craig, Gouveia and Haviar (2015). These digraphs, known as TiRS digraphs, have their origins in the dual representations of lattices by Urquhart (1978) and Ploščica (1995). We describe two properties of finite lattices which are weakenings of (upper) semimodularity and lower semimodularity respectively, and then show how these properties have a simple description in the dual digraphs. Combined with previous work on dual digraphs of semidistributive lattices (2022), it leads to a dual representation of finite meet-distributive lattices. This provides a natural link to finite convex geometries. In addition, we present two sufficient conditions on a finite TiRS digraph for its dual lattice to be modular. We close by posing four open problems.

math.RA↗

arXiv2021

Perfect extensions of de Morgan algebras

An algebra $\mathbb A$ is called a perfect extension of its subalgebra $\mathbb B$ if every congruence of $\mathbb B$ has a unique extension to $\mathbb A$. This terminology was used by Blyth and Varlet [1994]. In the case of lattices, this concept was described by Grätzer and Wehrung [1999] by saying that $\mathbb A$ is a congruence-preserving extension of $\mathbb B$. Not many investigations of this concept have been carried out so far. The present authors in another recent study faced the question of when a de Morgan algebra $\mathbb M$ is perfect extension of its Boolean subalgebra $B(\mathbb M)$, the so-called skeleton of $\mathbb M$. In this note a full solution to this interesting problem is given. The theory of natural dualities in the sense of Davey and Werner [1983] and Clark and Davey [1998], as well as Boolean product representations, are used as the main tools to obtain the solution.

math.LO↗

arXiv2020

Expanding Belnap 2: the dual category in depth

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' ($t$), `false'($f$), `contradiction' ($\top$) and `no information' ($\bot$), 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. In our companion paper, we introduced a new family of prioritised default bilattices, $\mathbf J_n$, for $n \in ω$, with $\mathbf J_0$ being Belnap's seminal example. We gave a duality for the variety $\mathcal V_n$ generated by $\mathbf J_n$, with the objects of the dual category $\mathcal X_n$ being multi-sorted topological structures. Here we study the dual category in depth. We give an axiomatisation of the category $\mathcal X_n$ and show that it is isomorphic to a category $\mathcal Y_n$ of single-sorted topological structures. The objects of $\mathcal Y_n$ are Priestley spaces endowed with a continuous retraction in which the order has a natural ranking. We show how to construct the Priestley dual of the underlying bounded distributive lattice of an algebra in $\mathcal V_n$ via its dual in $\mathcal Y_n$; as an application we show that the size of the free algebra $\mathbf F_{\mathcal V_n}(1)$ is given by a polynomial in $n$ of degree $6$.

math.LO↗

arXiv2020

Canonical extensions of lattices are more than perfect

In \cite{CGH15} we introduced TiRS graphs and TiRS frames to create a new natural setting for duals of canonical extensions of lattices. In this continuation of \cite{CGH15} we answer Problem 2 from there 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 introduce morphisms of TiRS structures and put our correspondence between TiRS graphs and TiRS frames from \cite{CGH15} into a full categorical framework. We illustrate our correspondences between classes of perfects lattices and classes of TiRS graphs by examples.

math.LO↗

arXiv2019

On selected developments in the theory of natural dualities

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. We present a brief background of the theory and then illustrate its applications on our study of Entailment Problem, Problem of Endodualisability versus Endoprimality and then a famous Full versus Strong Problem with related developments.

math.CT↗

arXiv2019

Congruence pairs of principal MS-algebras and perfect extensions

The notion of a congruence pair for principal MS-algebras, simpler than the one given by Beazer for $K_2$-algebras \cite{6}, is introduced. It is proved that the congruences of the principal MS-algebras $L$ correspond to the MS-congruence pairs on simpler substructures $L^{\circ\circ}$ and $D(L)$ of $L$ that were associated to~$L$ in \cite{4}. An analogy of a well-known Grätzer's problem \cite[Problem 57]{11} formulated for distributive p-algebras, which asks for a characterization of the congruence lattices in terms of the congruence pairs, is presented here for the principal MS-algebras (Problem 1). Unlike a recent solution to such a problem for the principal p-algebras in \cite{2}, it is demonstrated here on the class of principal MS-algebras, that a possible solution to the problem, though not very descriptive, can be simple and elegant. As a step to a more descriptive solution of Problem 1, a special case is then considered when a principal MS-algebra $L$ is a perfect extension of its greatest Stone subalgebra $L_{S}$. It is shown that this is exactly when de Morgan subalgebra $L^{\circ\circ}$ of $L$ is a perfect extension of the Boolean algebra $B(L)$. Two examples illustrating when this special case happens and when it does not are presented.

math.LO↗

arXiv2019

Expanding Belnap: dualities for a new class of default bilattices

Bilattices provide an algebraic tool with which to model simultaneously knowledge and truth. They were introduced by Belnap in 1977 in a paper entitled \emph{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' ($\top$) and `no information' ($\bot$). The resulting structure is equipped with two lattice orders, a \emph{knowledge order} and a \emph{truth order}, and hence is called a \emph{bilattice}. Prioritised default bilattices include not only values for `true' ($t_0$), `false' ($f_0$), `contradiction' and `no information', but also indexed families of default values, $t_1, \dots, t_n$ and $f_1, \dots, f_n$, for simultaneous modelling of degrees of knowledge and truth. We focus on a new family of prioritised default bilattices: $\mathbf J_n$, for $n \in ω$. The bilattice $\mathbf J_0$ 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 $\mathcal J_n$ generated by $\mathbf J_n$, and separately a multi-sorted topological representation for the bilattices in the variety $\mathcal V_n$ generated by $\mathbf J_n$. 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.

math.LO↗