Recent zbMATH articles in MSC 06Dhttps://zbmath.org/atom/cc/06D2024-04-02T17:33:48.828767ZWerkzeugBitopological duality for algebras of Fitting's logic and natural duality extensionhttps://zbmath.org/1529.031692024-04-02T17:33:48.828767Z"Das, Litan Kumar"https://zbmath.org/authors/?q=ai:das.litan-kumar"Ray, Kumar Sankar"https://zbmath.org/authors/?q=ai:ray.kumar-sankarSummary: In this paper, we investigate a bitopological duality for algebras of Fitting's multi-valued logic. We also extend the natural duality theory for \(\mathbb{ISP_I}(\mathcal{L})\) by developing a duality for \(\mathbb{ISP}(\mathcal{L})\), where \(\mathcal{L}\) is a finite algebra in which underlying lattice is bounded distributive.Falling shadow theory with applications in hoopshttps://zbmath.org/1529.032812024-04-02T17:33:48.828767Z"Borzooei, Rajab Ali"https://zbmath.org/authors/?q=ai:borzooei.rajab-ali"Rezaei, Gholam Reza"https://zbmath.org/authors/?q=ai:rezaei.gholamreza"Kologhani, Mona Aaly"https://zbmath.org/authors/?q=ai:kologhani.mona-aaly"Jun, Young Bae"https://zbmath.org/authors/?q=ai:jun.young-baeSummary: The falling shadow theory is applied to subhoops and filters in hoops. The notions of falling fuzzy subhoops and falling fuzzy filters in hoops are introduced, and several properties are investigated. Relationship between falling fuzzy subhoops and falling fuzzy filters are discussed, and conditions for a falling fuzzy subhoop to be a falling fuzzy filter are provided. Also conditions for a falling shadow of a random set to be a falling fuzzy filter are displayed.Quasi-Nelson algebras and fragmentshttps://zbmath.org/1529.032842024-04-02T17:33:48.828767Z"Rivieccio, Umberto"https://zbmath.org/authors/?q=ai:rivieccio.umberto"Jansana, Ramon"https://zbmath.org/authors/?q=ai:jansana.ramonSummary: The variety of quasi-Nelson algebras (QNAs) has been recently introduced and characterised in several equivalent ways: among others, as (1) the class of bounded commutative integral (but non-necessarily involutive) residuated lattices satisfying the Nelson identity, as well as (2) the class of (0, 1)-congruence orderable commutative integral residuated lattices. Logically, QNAs are the algebraic counterpart of quasi-Nelson logic, which is the (algebraisable) extension of the substructural logic \(\mathscr{F} \mathscr{L}_{\mathrm{ew}}\) (Full Lambek calculus with Exchange and Weakening) by the Nelson axiom. In the present paper, we collect virtually all the results that are currently known on QNAs, including solutions to certain questions left open in earlier publications. Furthermore, we extend our study to some subreducts of QNAs, that is, classes of algebras corresponding to fragments of the algebraic language obtained by eliding either the implication or the lattice operations.Some properties of lattice congruences preserving involutions and their largest numbers in the finite casehttps://zbmath.org/1529.060032024-04-02T17:33:48.828767Z"Mureşan, Claudia"https://zbmath.org/authors/?q=ai:muresan.claudiaSummary: In this paper, we characterize the congruences of an arbitrary i-lattice, investigate the structure of the lattice they form and how it relates to the structure of the lattice of lattice congruences, then, for an arbitrary nonzero natural number \(n\), we determine the largest possible number of congruences of an \(n\)-element i-lattice, along with the structures of the \(n\)-element i-lattices with this number of congruences. Our characterizations for the congruences of i-lattices have useful corollaries: a description of the atoms of the congruence lattices of i-lattices, congruence extensibility and characterization of subdirect irreducibility results. In terms of the relation between the above-mentioned problem on numbers of congruences of finite i-lattices and its analogue for lattices, while the \(n\)-element i-lattices with the largest number of congruences turn out to be exactly the \(n\)-element lattices whose number of congruences is either the largest or the second largest possible, we provide examples of pairs of \(n\)-element i-lattices and even pseudo-Kleene algebras such that one of them has strictly more congruences, but strictly less lattice congruences than the other.Canonical extensions of lattices are more than perfecthttps://zbmath.org/1529.060042024-04-02T17:33:48.828767Z"Craig, Andrew P. K."https://zbmath.org/authors/?q=ai:craig.andrew-p-k"Gouveia, Maria J."https://zbmath.org/authors/?q=ai:gouveia.maria-joao"Haviar, Miroslav"https://zbmath.org/authors/?q=ai:haviar.miroslavSummary: In a paper published in 2015 [Algebra Univers. 74, No. 1--2, 123--138 (2015; Zbl 1347.06008)], we introduced TiRS graphs and TiRS frames to create a new natural setting for duals of canonical extensions 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.A new approach on derivations of hyperlatticeshttps://zbmath.org/1529.060082024-04-02T17:33:48.828767Z"Koguep Njionou, Blaise Bleriot"https://zbmath.org/authors/?q=ai:koguep-njionou.blaise-bleriot"Chawoua Ngapeya, Audrey Armel"https://zbmath.org/authors/?q=ai:chawoua-ngapeya.audrey-armel"Leoreanu-Fotea, Violeta"https://zbmath.org/authors/?q=ai:leoreanu-fotea.violetaSummary: The main goal of this paper is to reintroduce and study the concept of derivation on hyperlattices. We introduce the notion of joinitive and strong-joinitive derivation and study some of their properties and relationship with isotone and contractive derivations on hyperlattices. Moreover, we study some properties of the fixed set of principal derivation and investigate the relationship between prime ideals and derivations in hyperlattice. Finally, we prove that the set of all principal derivations on a bounded hyperlattice \(\mathcal{L}\) is isomorphic to \(\mathcal{L}\) and we investigate the structure of the set of fixed set of principal derivations and established an isomorphism between this set and that of all the principal derivations on a distributive hyperlattice.Regular filters of distributive latticeshttps://zbmath.org/1529.060102024-04-02T17:33:48.828767Z"Sambasiva Rao, Mukkamala"https://zbmath.org/authors/?q=ai:rao.mukkamala-sambasiva"Phaneendra Kumar, Ananthapatnayakuni P."https://zbmath.org/authors/?q=ai:phaneendra-kumar.ananthapatnayakuni-pSummary: The concepts of regular filters and \(\pi\)-filters are introduced in distributive lattices. A set of equivalent conditions is given for a \(D\)-filter to become a regular filter. For every \(D\)-filter, it is proved that there exists a homomorphism whose dense kernel is a regular filter. \(\pi\)-filters are characterized in terms of regular filters and congruences. Some equivalent conditions are given for the space of all prime \(\pi\)-filters to become a Hausdorff space.Rings of quotients of the ring \(\mathcal{R}L\)https://zbmath.org/1529.060112024-04-02T17:33:48.828767Z"Abedi, Mostafa"https://zbmath.org/authors/?q=ai:abedi.mostafaSummary: Let \(\mathcal{Q}(\mathcal{R}L)\) (resp. \(\mathcal{Q}_{cl}(\mathcal{R}L)\)) denote the maximal ring of quotients (resp. classical ring of quotients) of the ring \(\mathcal{R}L\) of real-valued continuous functions on a completely regular frame \(L\). Then \(\mathcal{Q}(\mathcal{R}L) (\mathcal{Q}_{cl}(\mathcal{R}L))\) may be realized as the direct limit of the subrings \(\mathcal{R}(\downarrow c)\), where \(c\) is a dense element (dense cozero element) of \(L\). These representations of \(\mathcal{Q}(\mathcal{R}L)\) and \(\mathcal{Q}_{cl}(\mathcal{R}L)\) are applied to describe equalities among various rings of quotients
of \(\mathcal{R}L\). For example, we show that every dense open quotient of \(L\) is pseudocompact (that is, \(\mathcal{Q}(\mathcal{R}L) = \mathcal{Q}^* (\mathcal{R}L))\) if and only if \(L\) is a pseudocompact
\(P\)-frame.Banaschewski-Mulvey type compactification of proximal Császár frameshttps://zbmath.org/1529.060122024-04-02T17:33:48.828767Z"Shikweni, Pinkie"https://zbmath.org/authors/?q=ai:shikweni.pinkie"Nsonde-Nsayi, Jissy"https://zbmath.org/authors/?q=ai:nsayi.jissy-nsonde"Siweya, Hlengani"https://zbmath.org/authors/?q=ai:siweya.hlengani-jamesSummary: A Császár frame is said to be proximal if it is symmetric, strong and regular. Our aim in this paper is to apply the methods used by \textit{B. Banaschewski} and \textit{C. J. Mulvey} [J. Pure Appl. Algebra 33, 107--122 (1984; Zbl 0549.54017)] in constructing the Stone-Céch compactification of completely regular locale to construct a compactification of a proximal Császár frame.A note on \(3 \times 3\)-valued Łukasiewicz algebras with negationhttps://zbmath.org/1529.060132024-04-02T17:33:48.828767Z"Gallardo, Carlos"https://zbmath.org/authors/?q=ai:gallardo.carlos"Ziliani, Alicia"https://zbmath.org/authors/?q=ai:ziliani.aliciaSummary: In 2004, C. Sanza, with the purpose of legitimizing the study of \(n\times m\)-valued Łukasiewicz algebras with negation (or \(NS_{n\times m}\)-algebras) introduced \(3 \times 3\)-valued Łukasiewicz algebras with negation. Despite the various results obtained about \(NS_{n\times m}\)-algebras, the structure of the free algebras for this variety has not been determined yet. She only obtained a bound for their cardinal number with a finite number of free generators. In this note we describe the structure of the free finitely generated \(NS_{3 \times 3}\)-algebras and we determine a formula to calculate its cardinal number in terms of the number of free generators. Moreover, we obtain the lattice \(\Lambda(NS_{3\times 3})\) of all subvarieties of \(NS_{3\times 3}\) and we show that the varieties of Boolean algebras, three-valued Łukasiewicz algebras and four-valued Łukasiewicz algebras are proper subvarieties of \(NS_{3\times 3}\).Closure fuzzy filters of decomposable MS-algebrashttps://zbmath.org/1529.060142024-04-02T17:33:48.828767Z"Gubena, Yeshiwas Mebrat"https://zbmath.org/authors/?q=ai:gubena.yeshiwas-mebrat"Alemayehu, Teferi Getachew"https://zbmath.org/authors/?q=ai:alemayehu.teferi-getachew"Wondifraw, Yohannes Gedamu"https://zbmath.org/authors/?q=ai:wondifraw.yohannes-gedamuSummary: In this paper, we give some results of closure fuzzy filters of decomposable MS-algebras, characterization of closure fuzzy filters, and homomorphism of closure fuzzy filters. It is observed that the class of all closure fuzzy filters of a decomposable MS-algebra forms a bounded distributive lattice. We have also characterized the class of all closure fuzzy filters of a decopmposable MS-algebra in terms of boosters. Some properties of the homomorphic images and the inverse images of closure fuzzy filters are studied.Simple and subdirectly irreducible tense \(LM_\theta\)-algebrashttps://zbmath.org/1529.060152024-04-02T17:33:48.828767Z"Pascual, Inés"https://zbmath.org/authors/?q=ai:pascual.ines-bSummary: The main result of this paper is the determination of simple and subdirectly irreducible tense \(\theta\)-valued Łukasiewicz-Moisil algebras (or tense \(LM_\theta\)-algebras). To reach this aim, we use the topological duality for these algebras obtained by \textit{A. V. Figallo} et al. [Soft Comput. 23, No. 12, 3979--3997 (2019; Zbl 1418.03185)] and also the filters of these algebras. Besides, we define \(\theta\)-simple and \(\theta\)-subdirectly irreducible tense \(LM_\theta\)-algebras and we describe them. Moreover, we establish a relationship between simple and subdirectly irreducible tense \(LM_\theta\)-algebras and \(\theta\)-simple and \(\theta\)-subdirectly irreducible tense \(LM_\theta\)-algebras, respectively. Finally, we get a characterization of the subdirectly irreducible algebras which are complete and completely chrysippian tense \(LM_\theta\)-algebras and whose \(\theta\)-filters are complete.A finite axiomatization of positive MV-algebrashttps://zbmath.org/1529.060162024-04-02T17:33:48.828767Z"Abbadini, Marco"https://zbmath.org/authors/?q=ai:abbadini.marco"Jipsen, Peter"https://zbmath.org/authors/?q=ai:jipsen.peter"Kroupa, Tomáš"https://zbmath.org/authors/?q=ai:kroupa.tomas"Vannucci, Sara"https://zbmath.org/authors/?q=ai:vannucci.saraSummary: Positive MV-algebras are the subreducts of MV-algebras with respect to the signature \(\{\oplus, \odot, \vee, \wedge, 0, 1\}\). We provide a finite quasi-equational axiomatization for the class of such algebras.Primely filters in BL-algebrashttps://zbmath.org/1529.060172024-04-02T17:33:48.828767Z"Behzadi, Afsaneh"https://zbmath.org/authors/?q=ai:behzadi.afsaneh"Motamed, Somayeh"https://zbmath.org/authors/?q=ai:motamed.somayeh"Haghani, Farhad Khaksar"https://zbmath.org/authors/?q=ai:haghani.farhad-khaksarSummary: In this paper, we introduce the concept of primely filters in BL-algebras. As for the concept, we present some related results in BL-algebras. In particular, we show some relations between primely filters and other types of filters in BL-algebras.On weak convex MV-algebrashttps://zbmath.org/1529.060182024-04-02T17:33:48.828767Z"Dong, Yanyan"https://zbmath.org/authors/?q=ai:dong.yanyan"Shi, Fu-Gui"https://zbmath.org/authors/?q=ai:shi.fu-guiSummary: The aim of this paper is to introduce convex structures on MV-algebras such that the MV-operations are convexity preserving or weak convexity preserving. Therefore, we propose the concepts of paraconvex MV-algebras and weak convex MV-algebras. We give some characterizations of weak convex MV-algebras. Further, we show that the standard MV-algebra endowed with its interval convexity is a weak convex MV-algebra. In particular, a finite MV-chain endowed with a non-trivial convex structure is a weak convex MV-algebra iff the convex structure is precisely its interval convexity. Moreover, the direct product of finite weak convex MV-algebras is still a weak convex MV-algebra. Based on this, we further get that each finite MV-algebra endowed with its interval convexity is a weak convex MV-algebra. By using ideals, we introduce the ideal convexity on an MV-algebra which turns it to be a paraconvex MV-algebra. Finally, we discuss the separation axioms on weak convex MV-algebras.Reflectional topology in MV-algebrashttps://zbmath.org/1529.060192024-04-02T17:33:48.828767Z"Forouzesh, Fereshteh"https://zbmath.org/authors/?q=ai:forouzesh.fereshteh"Hosseini, Naser"https://zbmath.org/authors/?q=ai:hosseini.naserSummary: In this paper, we define soaker ideals in an MV-algebra, and study the relationships between soaker ideals and the other ideals in an involutive MV-algebras. Then we introduce a topology on the set of all the soaker ideals, which we call reflectional topology, and give a basis for it. By defining the notion of join-soaker ideals, we show that the reflectional topology is compact. We also give a characterization of connectedness of the reflectional topology. Finally, we investigate the properties of \(T_0\) and \(T_1\)-space in this topology.Radically principal MV-algebrashttps://zbmath.org/1529.060202024-04-02T17:33:48.828767Z"Heubo-Kwegna, Olivier A."https://zbmath.org/authors/?q=ai:heubo-kwegna.olivier-a"Nganou, Jean B."https://zbmath.org/authors/?q=ai:nganou.jean-bernardSummary: An MV-algebra \(A\) is radically principal if every prime ideal \(P\) of \(A\) is radically principal, i.e., there exists a principal ideal \(I\) of \(A\) such that \(\mathrm{Rad}(P)=\mathrm{Rad}(I)\). We investigate radically principal MV-algebras and provide some characterizations as well as some classes of examples. We prove a Cohen-like theorem, precisely, an MV-algebra is radically principal if and only if every maximal ideal is radically principal. It is also shown that the radically principal hyperarchemedian MV-algebras are the weakly finite ones and the radically principal Boolean algebras are the finite ones. Radically principal MV-algebras are also studied from the perspective of lattice-ordered groups.Weakly prime and super-max filters in BL-algebrashttps://zbmath.org/1529.060212024-04-02T17:33:48.828767Z"Moghaderi, Javad"https://zbmath.org/authors/?q=ai:moghaderi.javad"Motamed, Somayeh"https://zbmath.org/authors/?q=ai:motamed.somayehSummary: In this paper, the concepts of weakly prime filters and super-max filters in BL-algebras are introduced, and the relationships between them are discussed. Also, some properties and relations between these filters and other types of filters in BL-algebras are given. With some examples, it is shown that these filters have differences. After that, the notions of weakly linear BL-algebras and weak top BL-algebras are defined and investigated. Finally, using the notion of a weakly prime filter, a new topology on BL-algebras is defined and studied.Tense operators on BL-algebras and their applicationshttps://zbmath.org/1529.060222024-04-02T17:33:48.828767Z"Paad, Akbar"https://zbmath.org/authors/?q=ai:paad.akbarSummary: In this paper, the notions of tense operators and tense filters in BL-algebras are introduced and several characterizations of them are obtained. Also, the relation among tense BL-algebras, tense MV-algebras and tense Boolean algebras are investigated. Moreover, it is shown that the set of all tense filters of a BL-algebra is complete sublattice of \(F(L)\) of all filters of BL-algebra \(L\). Also, maximal tense filters and simple tense BL-algebras and the relation between them are studied. Finally, the notions of tense congruence relations in tense BL-algebras and strict tense BL-algebras are introduced and an one-to-one correspondence between tense filters and tense congruences relations induced by tense filters are provided.On inverse limit of a projective system of BL-algebrashttps://zbmath.org/1529.060232024-04-02T17:33:48.828767Z"Tayebi Khorami, Reza"https://zbmath.org/authors/?q=ai:tayebi-khorami.rezaSummary: In this paper, the inverse limits of a projective system of basic logic algebras (BL-algebras) are introduced, and their basic properties are studied. The set of congruences of a BL-algebra is considered as a poset. Then, a quotient inverse system and a quotient inverse limit on it are constructed. Moreover, by setting filters of a BL-algebra, quotient projective and inverse systems are constructed.An algebraic study of the logic \textbf{S5'(BL)}https://zbmath.org/1529.060242024-04-02T17:33:48.828767Z"Wang, Juntao"https://zbmath.org/authors/?q=ai:wang.juntao"He, Xiaoli"https://zbmath.org/authors/?q=ai:he.xiaoli"Wang, Mei"https://zbmath.org/authors/?q=ai:wang.mei.2Summary: \textit{P. Hájek} [Metamathematics of fuzzy logic. Dordrecht: Kluwer Academic Publishers (1998; Zbl 0937.03030)] introduced an S5-like modal fuzzy logic \textbf{S5(BL)} and showed that is equivalent to the monadic basic predicate logic \(\mathbf{mBL}_\forall\). Inspired by the above important results, D. Castaño et al. introduced monadic BL-algebras and their corresponding propositional logic \textbf{S5'(BL)}, which is a simplified set of axioms of \textbf{S5(BL)}. In this paper, we review the algebraic semantics of \textbf{S5'(BL)} and obtain some new results regarding to monadic BL-algebras. First we recall that \textbf{S5'(BL)} is completeness with respect to the variety \(\mathbb{MBL}\) of monadic BL-algebras and obtain a necessary and sufficient condition for the logic \textbf{S5'(BL)} to be semilinear. Then we study some further algebraic properties of monadic BL-algebras and discuss the relationship between monadic MV-algebras and monadic BL-algebras. Finally we give some characterizations of representable, simple, semisimple and directly indecomposable monadic BL-algebras, which are important members of the variety \(\mathbb{MBL}\). These results also constitute a crucial first step for providing an equivalent algebraic foundation for \(\mathbf{mBL}_\forall\).On the subalgebra lattice of a restricted Lie algebrahttps://zbmath.org/1529.170322024-04-02T17:33:48.828767Z"Páez-Guillán, Pilar"https://zbmath.org/authors/?q=ai:paez-guillan.pilar"Siciliano, Salvatore"https://zbmath.org/authors/?q=ai:siciliano.salvatore"Towers, David A."https://zbmath.org/authors/?q=ai:towers.david-a|towers.david-anthonyThe authors study the lattice of restricted subalgebras of a restricted Lie algebra. In particular, they consider those algebras in which this lattice is dually atomistic, lower or upper semimodular, or in which every restricted subalgebra is a quasi-ideal. \par The definitions of the notions above are as follows. Let \(L\) be a restricted Lie algebra. For a subset \(S\) of \(L\), denote by \(\langle S\rangle_p\) the restricted subalgebra generated by \(S\). 1) A restricted Lie algebra \(L\) is called \textit{dually atomistic} if every restricted subalgebra of \(L\) is an intersection of maximal restricted subalgebras of \(L\). 2) A restricted subalgebra \(U\) of \(L\) is called \textit{lower semimodular} in \(L\) if \(U \cap B\) is maximal in \(B\) for every restricted subalgebra \(B\) of \(L\) such that \(U\) is maximal in \(\langle U, B\rangle _p\). Also, \(L\) is said \textit{lower semimodular} if every restricted subalgebra of \(L\) is lower semimodular in \(L\). 3) A restricted subalgebra \(S\) of \(L\) is \textit{upper semimodular} in \(L\) if \(S\) is maximal in \(\langle S, T\rangle_p\) for every restricted subalgebra \(T\) of \(L\) such that \(S \cap T\) is maximal in \(T\). The restricted Lie algebra \(L\) is called \textit{upper semimodular} if all of its restricted subalgebras are upper semimodular in L. 4) A restricted subalgebra \(S\) of \(L\) is called a \textit{restricted quasi-ideal} of \(L\) if \([S, H] \subseteq S + H\) for all restricted subalgebras \(H\) of \(L\).
Reviewer: Victor Petrogradsky (Brasília)A new proof of the Joyal-Tierney theoremhttps://zbmath.org/1529.180082024-04-02T17:33:48.828767Z"Bezhanishvili, Guram"https://zbmath.org/authors/?q=ai:bezhanishvili.guram"Carai, Luca"https://zbmath.org/authors/?q=ai:carai.luca"Morandi, Patrick J."https://zbmath.org/authors/?q=ai:morandi.patrick-jA well-known result in locale theory, known as the Joyal-Tierney Theorem, states that a localic map \(f; M\rightarrow L\) is open iff its left adjoint is a complete Heyting homomorphism. In addition, if \(L\) is subfit, then \(f\) is open iff its left adjoint is a complete lattice homomorphism.
Priestley duality establishes a dual equivalence between the categories of bounded distributive lattices and Priestley spaces (remember that a Priestley space is a Stone spaces \(X\) equipped with a partial order \(\leqslant\) such that \(x \leqslant y\) implies the existence of a clopen upset \(U\) such that \(x\in U\) and \(y\notin U\)). \textit{A. Pultr} and \textit{J. Sichler} [Cah. Topologie Géom. Différ. Catégoriques 29, No. 3, 193--202 (1988; Zbl 0666.54018)] showed how to restrict Priestley duality to the category of frames.
In this paper, the authors aim is to give an alternative proof of this result using Priestley duality for frames. For this they need to translate the algebraic conditions of the Joyal-Tierney Theorem into geometric conditions about Priestley spaces.
As was shown in [\textit{J. Picado} and \textit{A. Pultr}, Frames and locales. Topology without points. Berlin: Springer (2012; Zbl 1231.06018)], if in the Joyal-Tierney Theorem it is asummend that \(L\) is subfit, then the localic map \(r : M\rightarrow L\) is open iff its left adjoint \( h: L\rightarrow M\) is a complete lattice homomorphism. The authors shall give an alternative proof of this result in the language of Priestle.
Reviewer: Joaquín Luna-Torres (Cartagena)Towards distributivity in FCA for phylogenetic datahttps://zbmath.org/1529.682952024-04-02T17:33:48.828767Z"Gély, Alain"https://zbmath.org/authors/?q=ai:gely.alain"Couceiro, Miguel"https://zbmath.org/authors/?q=ai:couceiro.miguel"Napoli, Amedeo"https://zbmath.org/authors/?q=ai:napoli.amedeoSummary: It is known that a distributive lattice is a median graph, and that a distributive \(\vee\)-semilattice can be thought of as a median graph iff every triple of elements such that the infimum of each couple of its elements exists, has an infimum. Since a lattice without its bottom element is obviously a \(\vee\)-semilattice, using the FCA formalism, we investigate the following problem: Given a semilattice \(L\) obtained from a lattice by deletion of the bottom element, is there a minimum distributive \(\vee\)-semilattice \(L_{d}\) such that \(L\) can be order embedded into \(L_{d}\)? We give a negative answer to this question by providing a counter-example.
For the entire collection see [Zbl 1491.68021].