New Class Of Algebras Research Articles

A fragment of a theoretical set and its strongly minimal central type

The paper defines a new class of algebras, the theory of which is a special case of Jonsson theories. This class applies to both varieties and Jonsson theories. The main results of this article are the following two results. In this article, an answer is obtained to the question of the equivalence of existential closure and algebraic closure of the model of the cosemantic class of a fixed spectrum of a Robinson hereditary variety. A criterion for strong minimality is obtained in the framework of the study of central types of central classes and fragments of a fixed spectrum.

Intuitionistic Modal Algebras

Recent research on algebraic models of quasi-Nelson logic has brought new attention to a number of classes of algebras which result from enriching (subreducts of) Heyting algebras with a special modal operator, known in the literature as a nucleus. Among these various algebraic structures, for which we employ the umbrella term intuitionistic modal algebras, some have been studied since at least the 1970s, usually within the framework of topology and sheaf theory. Others may seem more exotic, for their primitive operations arise from algebraic terms of the intuitionistic modal language which have not been previously considered. We shall for instance investigate the variety of weak implicative semilattices, whose members are (non-necessarily distributive) meet semilattices endowed with a nucleus and an implication operation which is not a relative pseudo-complement but satisfies the postulates of Celani and Jansana’s strict implication. For each of these new classes of algebras we establish a representation and a topological duality which generalize the known ones for Heyting algebras enriched with a nucleus.

Some Results about Weak UP‐algebras

We introduce a new class of algebras, weak UP‐algebras, briefly written as WUP‐algebras, as a wider class than the classes of UP‐algebras and KU‐algebras. We investigate fundamental aspects including maximal elements, branches and the subalgebra consisting of maximal elements of a WUP‐algebra. We also study regular congruences on a WUP‐algebra as well as the corresponding quotient WUP‐algebras. We prove that the congruence generated from the branches of a WUP‐algebra is regular and the corresponding quotient algebra χ/q is isomorphic to the WUP‐algebra Max(χ), the subalgebra of χ consisting of its maximal elements.

Some new criteria for small Igusa-Todorov algebras

n-Igusa-Todorov algebras are closely related to the finitistic dimension conjecture. Each n-Igusa-Todorov algebra has finite finitistic dimension. In this article, we find some new criteria for algebras being n-Igusa-Todorov algebras (here n = 0, 1). As applications, we extend the proof of the finitistic dimension conjecture to several new classes of algebras.

Quantum generalized Heisenberg algebras and their representations

We introduce and study a new class of algebras, which we name quantum generalized Heisenberg algebras (qGHA), including both the so-called generalized Heisenberg algebras and the generalized down-up algebras, but allowing more parameters of freedom, so as to encompass a wider range of applications and provide a common framework for several previously studied classes of algebras. In particular, our class includes the enveloping algebras of the Lie algebra and of the 3-dimensional Heisenberg Lie algebra, as well as its q-deformation, neither of which can be realized as a generalized Heisenberg algebra. This paper focuses mostly on the classification of finite-dimensional irreducible representations of qGHA, a study which reveals their rich structure. Although these algebras are not in general noetherian, their representations still retain a Lie-theoretic flavor. We work over a field of arbitrary characteristic and our results are presented in a characteristic-free fashion.

Structure and isomorphisms of quantum generalized Heisenberg algebras

In [S. A. Lopes and F. Razavinia, Quantum generalized Heisenberg algebras and their representations, preprint (2020), arXiv:2004.09301] we introduced a new class of algebras, which we named quantum generalized Heisenberg algebras and which depend on a parameter [Formula: see text] and two polynomials [Formula: see text]. We have shown that this class includes all generalized Heisenberg algebras (as defined in [E. M. F. Curado and M. A. Rego-Monteiro, Multi-parametric deformed Heisenberg algebras: A route to complexity, J. Phys. A: Math. Gen. 34(15) (2001) 3253; R. Lü and K. Zhao, Finite-dimensional simple modules over generalized Heisenberg algebras, Linear Algebra Appl. 475 (2015) 276–291, MR 3325233]) as well as generalized down-up algebras (as defined in [G. Benkart and T. Roby, Down-up algebras, J. Algebra 209(1) (1998) 305–344; T. Cassidy and B. Shelton, Basic properties of generalized down-up algebras, J. Algebra 279(1) (2004) 402–421, MR 2078408 (2005f:16051)]), but the parameters of freedom we allow for give rise to many algebras which are in neither one of these two classes. Having classified their finite-dimensional irreducible representations in [S. A. Lopes and F. Razavinia, Quantum generalized Heisenberg algebras and their representations, preprint (2020), arXiv:2004.09301], in this paper, we turn to their classification by isomorphism, the description of their automorphism groups and the study of their ring-theoretical properties.

Harish–Chandra modules over invariant subalgebras in a skew-group ring

We construct a new class of algebras resembling enveloping algebras and generalizing orthogonal Gelfand-Zeitlin algebras and rational Galois algebras studied by [EMV,FuZ,RZ,Har]. The algebras are defined via a geometric realization in terms of sheaves of functions invariant under an action of a finite group. A natural class of modules over these algebra can be constructed via a similar geometric realization. In the special case of a local reflection group, these modules are shown to have an explicit basis, generalizing similar results for orthogonal Gelfand-Zeitlin algebras from [EMV] and for rational Galois algebras from [FuZ]. We also construct a family of canonical simple Harish-Chandra modules and give sufficient conditions for simplicity of some modules.

New Class of Algebraic Fuzzy Systems Using Cubic Soft Sets with their Applications

In this paper, we stratify the connotation of cubic soft sets to ρ– algebras, and introduce the new class of cubic soft algebras like (e – CSp – SA) and (CSp – SA). We show that the R-union of two cubic soft ρ– sualgebras might not be cubic soft ρ– sualgebra. Furthermore, we show the sufficient condition to satisfy that the R-union of two cubic soft ρ– sualgebras is cubic soft ρ– sualgebra. Moreover, some of their basic characteristics are given.

Quasi-Nelson Algebras

We introduce a generalization of Nelson algebras having a not-necessarily involutive negation; we suggest to dub this class quasi-Nelson algebras in analogy with quasi-De Morgan lattices, these being a non-involutive generalization of De Morgan lattices. We show that, similarly to the involutive case (and perhaps surprisingly), our new class of algebras can be equivalently presented as (1) quasi-Nelson residuated lattices, i.e. models of the well-known Full Lambek calculus with exchange and weakening, extended with the Nelson axiom; (2) non-involutive twist-structures, i.e. special products of Heyting algebras, which generalize the well-known construction for representing algebraic models of Nelson's constructive logic with strong negation; (3) quasi-Nelson algebras, i.e. models of non-involutive Nelson logic viewed as a conservative expansion of the negation-free fragment of intuitionistic logic. The equivalence of the three presentations, and in particular the extension of the twist-structure representation to the non-involutive case, is the main technical result of the paper. We hope, however, that the main impact may be the possibility of opening new ways to (i) obtain deeper insights into the distinguishing feature of Nelson's logic (the Nelson axiom) and its algebraic counterpart; (ii) be able to investigate certain purely algebraic properties (such as 3-potency and (0,1)-congruence orderability) in a more general setting.

Degenerate 0-Schur Algebras and Nil-Temperley-Lieb Algebras

In Jensen and Su (J. Pure Appl. Algebra 219(2), 277–307 2014) constructed 0-Schur algebras, using double flag varieties. The construction leads to a presentation of 0-Schur algebras using quivers with relations and the quiver presentation naturally gives rise to a new class of algebras, which are introduced and studied in this paper. That is, these algebras are defined on the quivers of 0-Schur algebras with relations modified from the defining relations of 0-Schur algebras by a tuple of parameters t. In particular, when all the entries of t are 1, we recover 0-Schur algebras. When all the entries of t are zero, we obtain a class of basic algebras, which we call the degenerate 0-Schur algebras. We prove that the degenerate algebras are both associated graded algebras and quotients of 0-Schur algebras. Moreover, we give a geometric interpretation of the degenerate algebras using double flag varieties, in the same spirit as Jensen and Su (J. Pure Appl. Algebra 219(2), 277–307 2014), and show how the centralizer algebras are related to nil-Hecke and nil-Temperley-Lieb algebras.

Generalized Knot Symmetric Algebras

Attempts have been made to introduce a new class of algebras $$K_n$$, which we define as generalized knot symmetric algebras. The reason for this name is that these new algebras have a basis consisting of knot diagrams. The multiplication of two of these graphs turns $$K_n$$ into an associative. The semi-simplicity of these algebras over $$K_n$$ has been proved by making use of conditional exception and proving the non-degeneracy of the trace.

Ringel duality and Auslander–Dlab–Ringel algebras

We introduce a new class of quasi-hereditary algebras, containing in particular the Auslander–Dlab–Ringel (ADR) algebras. We show that this new class of algebras is preserved under Ringel duality, which determines in particular explicitly the Ringel dual of any ADR algebra. As a special case of our theory, it follows that, under very restrictive conditions, an ADR algebra is Ringel dual to another one. The latter provides an alternative proof for a recent result of Conde and Erdmann, and places it in a more general setting.

Morphisms on $${ EMV}$$ EMV -algebras and their applications

For a new class of algebras, called \({ EMV}\)-algebras, every idempotent element a determines an \({ MV}\)-algebra which is important for the structure of the \({ EMV}\)-algebra. Therefore, instead of standard homomorphisms of \({ EMV}\)-algebras, we introduce \({ EMV}\)-morphisms as a family of \({ MV}\)-homomorphisms from \({ MV}\)-algebras [0, a] into other ones. \({ EMV}\)-morphisms enable us to study categories of \({ EMV}\)-algebras where objects are \({ EMV}\)-algebras and morphisms are special classes of \({ EMV}\)-morphisms. The category is closed under product. In addition, we define free \({ EMV}\)-algebras on a set X with respect to \({ EMV}\)-morphisms. If X is finite, then a free \({ EMV}\)-algebra on X is termwise equivalent to the free \({ MV}\)-algebra on X. For an infinite set X, the same is true introducing a so-called weakly free \({ EMV}\)-algebra.

Introducing fully UP-semigroups

In this paper, we introduce some new classes of algebras related to UP-algebras and semigroups, called a left UP-semigroup, a right UP-semigroup, a fully UP-semigroup, a left-left UP-semigroup, a right-left UP-semigroup, a left-right UP-semigroup, a right-right UP-semigroup, a fully-left UP-semigroup, a fully-right UP-semigroup, a left-fully UP-semigroup, a right-fully UP-semigroup, a fully-fully UP-semigroup, and find their examples.

Closed ideals in a new class of algebras of holomorphic functions on the disc

We define a new class of Banach algebras of holomorphic functions on the unit disc \(\mathbb{D}\) which contains the algebras studied in [GMR2] and [GW]. To a function G of the class \(\mathcal{A}^1 (\mathbb{D})\) nowhere vanishing in \(\mathbb{D}\) we associate a Banach algebra \(\mathcal{A}^n_G (\mathbb{D})\) contained in the disc algebra \(\mathcal{A}(\mathbb{D})\). We prove that all closed ideals of \(\mathcal{A}^n_G (\mathbb{D})\) of at most countable hull are of the standard form.

On the Intersection Algebra of Principal Ideals

We study the finite generation of the intersection algebra of two principal ideals I and J in a unique factorization domain R. We provide an algorithm that produces a list of generators of this algebra over R. In the special case that R is a polynomial ring, this algorithm has been implemented in the commutative algebra software system Macaulay2. A new class of algebras, called fan algebras, is introduced.

Bilattices with Implications

In a previous work we studied, from the perspective of Abstract Algebraic Logic, the implicationless fragment of a logic introduced by O. Arieli and A. Avron using a class of bilattice-based logical matrices called logical bilattices. Here we complete this study by considering the Arieli-Avron logic in the full language, obtained by adding two implication connectives to the standard bilattice language. We prove that this logic is algebraizable and investigate its algebraic models, which turn out to be distributive bilattices with additional implication operations. We axiomatize and state several results on these new classes of algebras, in particular representation theorems analogue to the well-known one for interlaced bilattices.

Algebra of Rough Sets based on Quasi Order

In this paper, we give a characterization theorem for rough sets based on quasi order. We obtain an algebra on the rough sets system determined by a quasi order which is the generalization of the algebra of rough sets system determined by an equivalence relation given in [1]. The properties of this algebra are abstracted at various levels and define new class of algebras. Further we give a representation theorem for the new class of algebras.

Representations of quiver Hecke algebras via Lyndon bases

A new class of algebras has been introduced by Khovanov and Lauda and independently by Rouquier. These algebras categorify one-half of the Quantum group associated to arbitrary Cartan data. In this paper, we use the combinatorics of Lyndon words to construct the irreducible representations of those algebras associated to Cartan data of finite type. This completes the classification of simple modules for the quiver Hecke algebra initiated by Kleshchev and Ram.

Reconstruction algebras of type 𝐴

We introduce a new class of algebras, called reconstruction algebras, and present some of their basic properties. These non-commutative rings dictate in every way the process of resolving the Cohen-Macaulay singularities C 2 / G \mathbb {C}^2/G where G = 1 r ( 1 , a ) ≤ G L ( 2 , C ) G=\frac {1}{r}(1,a)\leq GL(2,\mathbb {C}) .