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The lattice of subvarieties of the biregularization of the variety of Boolean algebras

100%

Płonka J.

Discussiones Mathematicae - General Algebra and Applications

2001

tom 21

nr 2

255-268

Let τ: F → N be a type of algebras, where F is a set of fundamental operation symbols and N is the set of all positive integers. An identity φ ≈ ψ is called biregular if it has the same variables in each of it sides and it has the same fundamental operation symbols in each of it sides. For a variety V of type τ we denote by $V_{b}$ the biregularization of V, i.e. the variety of type τ defined by all biregular identities from Id(V). Let B be the variety of Boolean algebras of type $τ_{b}: {+,·,´} → N$, where $τ_{b}(+) = τ_{b}(·) = 2$ and $τ_{b}(´) = 1$. In this paper we characterize the lattice $ℒ(B_{b})$ of all subvarieties of the biregularization of the variety B.

A Note on 3×3-valued Łukasiewicz Algebras with Negation

100%

Gallardo C. , Ziliani A.

Bulletin of the Section of Logic

2021

tom 50

nr 3

289-298

In 2004, C. Sanza, with the purpose of legitimizing the study of $n\times m$-valued Łukasiewicz algebras with negation (or $\mathbf{NS}_{n\times m}$-algebras) introduced $3 \times 3$-valued Łukasiewicz algebras with negation. Despite the various results obtained about $\mathbf{NS}_{n\times m}$-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 $\Lambda(\mathbf{NS}_{3\times 3})$ of all subvarieties of $\mathbf{NS}_{3\times 3}$ and we show that the varieties of Boolean algebras, three-valued Łukasiewicz algebras and four-valued Łukasiewicz algebras are proper subvarieties of $\mathbf{NS}_{3\times 3}$.

Komori identities in algebraic logic

88%

Blok W. , La Falce S.

Reports on Mathematical Logic

2000

tom No. 34

79-106

A variety generated by a class K of BCK-algebras consists of BCK-algebras if and only if it satisfies a certain kind of identity, first discovered by Komori. A similar phenomenon is shown to hold more generally in a certain class of quasivarieties of logic that includes not only the class of BCK-algebras but also such classes as the quasivariety of biresiduation algebras and quasivarieties of algebras with an equivalence operation. We describe a set of identities (which we call Komori identities), and show that the variety generated by a class K of algebras in one of the quasivarieties considered is contained in the quasivariety if and only it it satisfies a Komori identity. We use the result to establish (i) that the subvarieties of any of the quasivarieties studied are congruence 3-permutable and (ii) that the varietal join of two subvarieties of any of the quasivarieties studied is contained in the quasivariety.

On varieties of left distributive left idempotent groupoids

75%

Stanovský D.

Discussiones Mathematicae - General Algebra and Applications

2004

tom 24

nr 2

267-275

We describe a part of the lattice of subvarieties of left distributive left idempotent groupoids (i.e. those satisfying the identities x(yz) ≈ (xy)(xz) and (xx)y ≈ xy) modulo the lattice of subvarieties of left distributive idempotent groupoids. A free groupoid in a subvariety of LDLI groupoids satisfying an identity xⁿ ≈ x decomposes as the direct product of its largest idempotent factor and a cycle. Some properties of subdirectly ireducible LDLI groupoids are found.

A note on regular De Morgan semi-Heyting algebras

63%

Sankappanavar H. P.

Demonstratio Mathematica

2016

tom Vol. 49, nr 3

252--256

The purpose of this note is two-fold. Firstly, we prove that the variety RDMSH1 of regular De Morgan semi-Heyting algebras of level 1 satisfies Stone identity and present (equational) axiomatizations for several subvarieties of RDMSH1. Secondly, using our earlier results published in 2014, we give a concrete description of the lattice of subvarieties of the variety RDQDStSH1 of regular dually quasi-De Morgan Stone semi-Heyting algebras that contains RDMSH1. Furthermore, we prove that every subvariety of RDQDStSH1, and hence of RDMSH1, has Amalgamation Property. The note concludes with some open problems for further investigation.