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Symmetry
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The Study of Lattice Theory and Universal Algebra
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The Study of Lattice Theory and Universal Algebra

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (17 February 2023) | Viewed by 13979

Special Issue Editor

Prof. Ivan Chajda

E-Mail Website
Guest Editor
Dept. Algebra & Geometry, Palacky Univrsity, 77146 Olomouc, Czech Republic
Interests: lattice theory; universal algebra

Special Issue Information

Dear Colleagues,

Lattice theory in the present conception was initiated by publishing Garrett Birkhoff’s seminal book in 1940. Since then, it has been an extensively developed branch which is still accepting new concepts, results, and applications. In its contemporary state, there are several important applications of lattice theory, e.g., in algebraic semantics of non-classical logics. Let us remember the works of Brouwer and Heyting on pseudocomplemented and relatively pseudocomplemented lattices and semilattices for algebraization of intuitionistic logic, orthomodular lattices, and posits for the formalization of the logic of quantum mechanics, an application of lattice theory for various algebraic axiomatizations of many-valued logics such as MV algebras or residuated lattices on which fuzzy logic and several substructural logics are based. Moreover, these applications and results are being developed quickly at present, and they also influence the general theory of lattices.

Universal algebra before G. Birkhoff’s famous papers in the 1930s was only a generalization of known results for groups, rings, semigroups, etc. However, Birkhoff’s results on a variety of algebras established a fundamental step toward an advanced theory which was excellently settled and developed by George Grätzer in his remarkable monograph Universal Algebra in 1968. This was a keystone for the next development of this part of algebra. A number of papers from this branch were published every year, bringing new and unexpected results and trends, e.g., the connection between universal algebra and lattice theory, which is an axis of these two branches, and the theory of algebraic duality or results on lattices of subvarieties of congruence distributive varieties.

We encourage researchers to submit papers addressing these topics to this Special Issue of the journal Symmetry. 

Prof. Ivan Chajda
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  •     general aspects of lattices, semilattices and ordered sets
  •     applications of lattice theory in non-classical logics
  •     residuated lattices
  •     relatively and sectionally pseudocomplemented lattices
  •     application of lattice theory in the logic of quantum mechanics
  •     orthomodular lattices, orthomodular posits, and their generalizations
  •     application of lattice theory in geometry
  •     algebraic theory of ordered sets
  •     general aspects of universal algebra
  •     congruence conditions
  •     varieties of algebras
  •     application of universal algebra

Published Papers (8 papers)

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Research

13 pages, 422 KiB  
Article
Orthomodular and Skew Orthomodular Posets
by Ivan Chajda, Miroslav Kolařík and Helmut Länger
Symmetry 2023, 15(4), 810; https://doi.org/10.3390/sym15040810 - 27 Mar 2023
Cited by 1 | Viewed by 1144
Abstract
We present the smallest non-lattice orthomodular poset and show that it is unique up to isomorphism. Since not every Boolean poset is orthomodular, we consider the class of skew orthomodular posets previously introduced by the first and third author under the name “generalized [...] Read more.
We present the smallest non-lattice orthomodular poset and show that it is unique up to isomorphism. Since not every Boolean poset is orthomodular, we consider the class of skew orthomodular posets previously introduced by the first and third author under the name “generalized orthomodular posets”. We show that this class contains all Boolean posets and we study its subclass consisting of horizontal sums of Boolean posets. For this purpose, we introduce the concept of a compatibility relation and the so-called commutator of two elements. We show the relationship between these concepts and introduce a kind of ternary discriminator for horizontal sums of Boolean posets. Numerous examples illuminating these concepts and results are included in the paper. Full article
(This article belongs to the Special Issue The Study of Lattice Theory and Universal Algebra)
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13 pages, 305 KiB  
Article
Esakia Duality for Heyting Small Spaces
by Artur Piękosz
Symmetry 2022, 14(12), 2567; https://doi.org/10.3390/sym14122567 - 5 Dec 2022
Cited by 1 | Viewed by 1148
Abstract
We continue our research plan of developing the theory of small and locally small spaces, proposing this theory as a realisation of Grothendieck’s idea of tame topology on the level of general topology. In this paper, we develop the theory of Heyting small [...] Read more.
We continue our research plan of developing the theory of small and locally small spaces, proposing this theory as a realisation of Grothendieck’s idea of tame topology on the level of general topology. In this paper, we develop the theory of Heyting small spaces and prove a new version of Esakia Duality for such spaces. To do this, we notice that spectral spaces may be seen as sober small spaces with all smops compact and introduce the method of the standard spectralification. This helps to understand open continuous definable mappings between definable spaces over o-minimal structures. Full article
(This article belongs to the Special Issue The Study of Lattice Theory and Universal Algebra)
10 pages, 962 KiB  
Article
Characterizations of Γ Rings in Terms of Rough Fuzzy Ideals
by Durgadevi Pushpanathan and Ezhilmaran Devarasan
Symmetry 2022, 14(8), 1705; https://doi.org/10.3390/sym14081705 - 16 Aug 2022
Cited by 2 | Viewed by 1077
Abstract
Fuzzy sets are a major simplification and wing of classical sets. The extended concept of set theory is rough set (RS) theory. It is a formalistic theory based upon a foundational study of the logical features of the fundamental system. The RS theory [...] Read more.
Fuzzy sets are a major simplification and wing of classical sets. The extended concept of set theory is rough set (RS) theory. It is a formalistic theory based upon a foundational study of the logical features of the fundamental system. The RS theory provides a new mathematical method for insufficient understanding. It enables the creation of sets of verdict rules from data in a presentable manner. An RS boundary area can be created using the algebraic operators union and intersection, which is known as an approximation. In terms of data uncertainty, fuzzy set theory and RS theory are both applicable. In general, as a uniting theme that unites diverse areas of modern arithmetic, symmetry is immensely important and helpful. The goal of this article is to present the notion of rough fuzzy ideals (RFI) in the gamma ring structure. We introduce the basic concept of RFI, and the theorems are proven for their characteristic function. After that, we explain the operations on RFI, and related theorems are given. Additionally, we prove some theorems on rough fuzzy prime ideals. Furthermore, using the concept of rough gamma endomorphism, we propose some theorems on the morphism properties of RFI in the gamma ring. Full article
(This article belongs to the Special Issue The Study of Lattice Theory and Universal Algebra)
22 pages, 549 KiB  
Article
Groups and Structures of Commutative Semigroups in the Context of Cubic Multi-Polar Structures
by Anas Al-Masarwah, Mohammed Alqahtani and Majdoleen Abu Qamar
Symmetry 2022, 14(7), 1493; https://doi.org/10.3390/sym14071493 - 21 Jul 2022
Viewed by 1385
Abstract
In recent years, the m-polar fuzziness structure and the cubic structure have piqued the interest of researchers and have been commonly implemented in algebraic structures like groupoids, semigroups, groups, rings and lattices. The cubic m-polar (CmP) structure [...] Read more.
In recent years, the m-polar fuzziness structure and the cubic structure have piqued the interest of researchers and have been commonly implemented in algebraic structures like groupoids, semigroups, groups, rings and lattices. The cubic m-polar (CmP) structure is a generalization of m-polar fuzziness and cubic structures. The intent of this research is to extend the CmP structures to the theory of groups and semigroups. In the present research, we preface the concept of the CmP groups and probe many of its characteristics. This concept allows the membership grade and non-membership grade sequence to have a set of m-tuple interval-valued real values and a set of m-tuple real values between zero and one. This new notation of group (semigroup) serves as a bridge among CmP structure, classical set and group (semigroup) theory and also shows the effect of the CmP structure on a group (semigroup) structure. Moreover, we derive some fundamental properties of CmP groups and support them by illustrative examples. Lastly, we vividly construct semigroup and groupoid structures by providing binary operations for the CmP structure and provide some dominant properties of these structures. Full article
(This article belongs to the Special Issue The Study of Lattice Theory and Universal Algebra)
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14 pages, 272 KiB  
Article
Bipolar Fuzzy Set Theory Applied to the Certain Ideals in BCI-Algebras
by N. Abughazalah, G. Muhiuddin, Mohamed E. A. Elnair and A. Mahboob
Symmetry 2022, 14(4), 815; https://doi.org/10.3390/sym14040815 - 14 Apr 2022
Cited by 5 | Viewed by 1584
Abstract
The study of symmetry is one of the most important and beautiful themes uniting various areas of contemporary arithmetic. Algebraic structures are useful structures in pure mathematics for learning a geometrical object’s symmetries. In this paper, we introduce new concepts in an algebraic [...] Read more.
The study of symmetry is one of the most important and beautiful themes uniting various areas of contemporary arithmetic. Algebraic structures are useful structures in pure mathematics for learning a geometrical object’s symmetries. In this paper, we introduce new concepts in an algebraic structure called BCI-algebra, where we present the concepts of bipolar fuzzy (closed) BCI-positive implicative ideals and bipolar fuzzy (closed) BCI-commutative ideals of BCI-algebras. The relationship between bipolar fuzzy (closed) BCI-positive implicative ideals and bipolar fuzzy ideals is investigated, and various conditions are provided for a bipolar fuzzy ideal to be a bipolar fuzzy BCI-positive implicative ideal. Furthermore, conditions are presented for a bipolar fuzzy (closed) ideal to be a bipolar fuzzy BCI-commutative ideal. Full article
(This article belongs to the Special Issue The Study of Lattice Theory and Universal Algebra)
40 pages, 651 KiB  
Article
Centralising Monoids with Low-Arity Witnesses on a Four-Element Set
by Mike Behrisch and Edith Vargas-García
Symmetry 2021, 13(8), 1471; https://doi.org/10.3390/sym13081471 - 11 Aug 2021
Cited by 3 | Viewed by 1337
Abstract
As part of a project to identify all maximal centralising monoids on a four-element set, we determine all centralising monoids witnessed by unary or by idempotent binary operations on a four-element set. Moreover, we show that every centralising monoid on a set with [...] Read more.
As part of a project to identify all maximal centralising monoids on a four-element set, we determine all centralising monoids witnessed by unary or by idempotent binary operations on a four-element set. Moreover, we show that every centralising monoid on a set with at least four elements witnessed by the Mal’cev operation of a Boolean group operation is always a maximal centralising monoid, i.e., a co-atom below the full transformation monoid. On the other hand, we also prove that centralising monoids witnessed by certain types of permutations or retractive operations can never be maximal. Full article
(This article belongs to the Special Issue The Study of Lattice Theory and Universal Algebra)
10 pages, 272 KiB  
Article
Identities Generalizing the Theorems of Pappus and Desargues
by Roger D. Maddux
Symmetry 2021, 13(8), 1382; https://doi.org/10.3390/sym13081382 - 29 Jul 2021
Viewed by 1637
Abstract
The Theorems of Pappus and Desargues (for the projective plane over a field) are generalized here by two identities involving determinants and cross products. These identities are proved to hold in the three-dimensional vector space over a field. They are closely related to [...] Read more.
The Theorems of Pappus and Desargues (for the projective plane over a field) are generalized here by two identities involving determinants and cross products. These identities are proved to hold in the three-dimensional vector space over a field. They are closely related to the Arguesian identity in lattice theory and to Cayley-Grassmann identities in invariant theory. Full article
(This article belongs to the Special Issue The Study of Lattice Theory and Universal Algebra)
13 pages, 288 KiB  
Article
Semigroups of Terms, Tree Languages, Menger Algebra of n-Ary Functions and Their Embedding Theorems
by Thodsaporn Kumduang and Sorasak Leeratanavalee
Symmetry 2021, 13(4), 558; https://doi.org/10.3390/sym13040558 - 27 Mar 2021
Cited by 16 | Viewed by 2027
Abstract
The concepts of terms and tree languages are significant tools for the development of research works in both universal algebra and theoretical computer science. In this paper, we establish a strong connection between semigroups of terms and tree languages, which provides the tools [...] Read more.
The concepts of terms and tree languages are significant tools for the development of research works in both universal algebra and theoretical computer science. In this paper, we establish a strong connection between semigroups of terms and tree languages, which provides the tools for studying monomorphisms between terms and generalized hypersubstitutions. A novel concept of a seminearring of non-deterministic generalized hypersubstitutions is introduced and some interesting properties among subsets of its are provided. Furthermore, we prove that there are monomorphisms from the power diagonal semigroup of tree languages and the monoid of generalized hypersubstitutions to the power diagonal semigroup of non-deterministic generalized hypersubstitutions and the monoid of non-deterministic generalized hypersubstitutions, respectively. Finally, the representation of terms using the theory of n-ary functions is defined. We then present the Cayley’s theorem for Menger algebra of terms, which allows us to provide a concrete example via full transformation semigroups. Full article
(This article belongs to the Special Issue The Study of Lattice Theory and Universal Algebra)
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