Free spectra of linear equivalential algebras

2005 ◽  
Vol 70 (4) ◽  
pp. 1341-1358 ◽  
Author(s):  
Katarzyna Slomczyńska
Keyword(s):  
Free Algebras ◽  
Finitely Generated ◽  
Dummett Logic ◽  
Finite Height ◽  
Free Spectra ◽  
Equivalential Algebras

AbstractWe construct the finitely generated free algebras and determine the free spectra of varieties of linear equivalential algebras and linear equivalential algebras of finite height corresponding, respectively, to the equivalential fragments of intermediate Gödel-Dummett logic and intermediate finite-valued logics of Gödel. Thus we compute the number of purely equivalential propositional formulas in these logics in n variables for an arbitrary n ∈ ℕ.

scholarly journals Equivalential algebras with conjunction on the regular elements

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Sławomir Przybyło
Keyword(s):  
Representation Theorem ◽  
Free Algebras ◽  
Finitely Generated ◽  
Regular Elements ◽  
Definition Of ◽  
Equivalential Algebra ◽  
Free Spectra ◽  
Equivalential Algebras

Abstract We introduce the definition of the three-element equivalential algebra R with conjunction on the regular elements. We study the variety generated by R and prove the Representation Theorem. Then, we construct the finitely generated free algebras and compute the free spectra in this variety.

Finitely generated free Heyting algebras

1986 ◽  
Vol 51 (1) ◽  
pp. 152-165 ◽  
Author(s):  
Fabio Bellissima
Keyword(s):  
Free Algebra ◽  
Kripke Semantics ◽  
Heyting Algebras ◽  
Free Algebras ◽  
Finitely Generated ◽  
Algebraic Properties

AbstractThe aim of this paper is to give, using the Kripke semantics for intuitionism, a representation of finitely generated free Heyting algebras. By means of the representation we determine in a constructive way some set of “special elements” of such algebras. Furthermore, we show that many algebraic properties which are satisfied by the free algebra on one generator are not satisfied by free algebras on more than one generator.

scholarly journals Computation of minimal homogeneous generating sets and minimal standard bases for ideals of free algebras

2018 ◽  
Vol 25 (3) ◽  
pp. 451-459
Author(s):  
Huishi Li
Keyword(s):  
Free Algebra ◽  
Standard Basis ◽  
Free Algebras ◽  
Grobner Basis ◽  
Finitely Generated ◽  
Generating Set ◽  
Standard Bases ◽  
Generating Sets ◽  
Monomial Ordering ◽  
Minimal Standard

AbstractLet {K\langle X\rangle=K\langle X_{1},\ldots,X_{n}\rangle} be the free algebra generated by {X=\{X_{1},\ldots,X_{n}\}} over a field K. It is shown that, with respect to any weighted {\mathbb{N}}-gradation attached to {K\langle X\rangle}, minimal homogeneous generating sets for finitely generated graded two-sided ideals of {K\langle X\rangle} can be algorithmically computed, and that if an ungraded two-sided ideal I of {K\langle X\rangle} has a finite Gröbner basis {{\mathcal{G}}} with respect to a graded monomial ordering on {K\langle X\rangle}, then a minimal standard basis for I can be computed via computing a minimal homogeneous generating set of the associated graded ideal {\langle\mathbf{LH}(I)\rangle}.

Types of points and algebras

2018 ◽  
Vol 28 (08) ◽  
pp. 1717-1730 ◽  
Author(s):  
G. Zhitomirski
Keyword(s):  
Classical Model ◽  
Free Algebras ◽  
Finitely Generated ◽  
Universal Algebras

The connection between classical model theoretical types (MT-types) and logically-geometrical types (LG-types) introduced by B. Plotkin is considered. It is proved that MT-types of two [Formula: see text]-tuples in two universal algebras coincide if and only if their LG-types coincide. Two problems set by B. Plotkin are considered: (1) let two tuples in an algebra have the same type, does it imply that they are connected by an automorphism of this algebra? and (2) let two algebras have the same type, does it imply that they are isomorphic? Some varieties of universal algebras are considered having in view these problems. In particular, it is proved that if a variety is hopfian or co-hopfian, then finitely generated free algebras of such a variety are completely determined by their type.

scholarly journals ON FREE SPECTRA OF LOCALLY TESTABLE SEMIGROUP VARIETIES

2011 ◽  
Vol 53 (3) ◽  
pp. 623-629 ◽  
Author(s):  
IGOR DOLINKA
Keyword(s):  
Asymptotic Behaviour ◽  
Finitely Generated ◽  
Free Objects ◽  
Free Spectra ◽  
Semigroup Varieties

AbstractFor each k ≥ 2, we determine the asymptotic behaviour of the sequence of cardinalities of finitely generated free objects in , the variety consisting of all k-testable semigroups.

scholarly journals Free Heyting algebra endomorphisms: Ruitenburg’s Theorem and beyond

2020 ◽  
Vol 30 (6) ◽  
pp. 572-596 ◽  
Author(s):  
Silvio Ghilardi ◽  
Luigi Santocanale
Keyword(s):  
Heyting Algebra ◽  
Free Algebras ◽  
Finitely Generated ◽  
Locally Finite ◽  
Duality Techniques

AbstractRuitenburg’s Theorem says that every endomorphism f of a finitely generated free Heyting algebra is ultimately periodic if f fixes all the generators but one. More precisely, there is N ≥ 0 such that fN+2 = fN, thus the period equals 2. We give a semantic proof of this theorem, using duality techniques and bounded bisimulation ranks. By the same techniques, we tackle investigation of arbitrary endomorphisms of free algebras. We show that they are not, in general, ultimately periodic. Yet, when they are (e.g. in the case of locally finite subvarieties), the period can be explicitly bounded as function of the cardinality of the set of generators.

Sugihara algebras and Sugihara monoids: Multisorted dualities

2020 ◽  
Vol 70 (1) ◽  
pp. 5-28
Author(s):  
Leonardo M. Cabrer ◽  
Hilary A. Priestley
Keyword(s):  
Duality Theory ◽  
Theoretical Framework ◽  
Deductive System ◽  
Relevance Logic ◽  
Free Algebras ◽  
Finitely Generated ◽  
Duality Theorems ◽  
Admissible Rules ◽  
Viable Method

AbstractThe authors developed in a recent paper natural dualities for finitely generated quasivarieties of Sugihara algebras. They thereby identified the admissibility algebras for these quasivarieties which, via the Test Spaces Method devised by Cabrer et al., give access to a viable method for studying admissible rules within relevance logic, specifically for extensions of the deductive system R-mingle.This paper builds on the work already done on the theory of natural dualities for Sugihara algebras. Its purpose is to provide an integrated suite of multisorted duality theorems of a uniform type, encompassing finitely generated quasivarieties and varieties of both Sugihara algebras and Sugihara monoids, and embracing both the odd and the even cases. The overarching theoretical framework of multisorted duality theory developed here leads on to amenable representations of free algebras. More widely, it provides a springboard to further applications.

scholarly journals AUTOMORPHISMS OF THE CATEGORY OF THE FREE NILPOTENT GROUPS OF THE FIXED CLASS OF NILPOTENCY

2007 ◽  
Vol 17 (05n06) ◽  
pp. 1273-1281 ◽  
Author(s):  
A. TSURKOV
Keyword(s):  
Algebraic Geometry ◽  
Nilpotent Groups ◽  
Nilpotency Class ◽  
Free Algebras ◽  
Finitely Generated ◽  
Identity Automorphism ◽  
Universal Algebraic Geometry ◽  
Inner Automorphisms

This paper is motivated by the following question arising in universal algebraic geometry: when do two algebras have the same geometry? This question requires considering algebras in a variety Θ and the category Θ0 of all finitely generated free algebras in Θ. The key problem is to study how far the group Aut Θ0 of all automorphisms of the category Θ0 is from the group Inn Θ0 of inner automorphisms of Θ0 (see [7, 10] for details). Recall that an automorphism ϒ of a category 𝔎 is inner, if it is isomorphic as a functor to the identity automorphism of the category 𝔎. Let Θ = 𝔑d be the variety of all nilpotent groups whose nilpotency class is ≤ d. Using the method of verbal operations developed in [8, 9], we prove that every automorphism of the category [Formula: see text], d ≥ 2 is inner.

scholarly journals Trees as commutative BCK-Algebras

1981 ◽  
Vol 23 (2) ◽  
pp. 181-190
Author(s):  
William H. Cornish
Keyword(s):  
Height Function ◽  
Chain Condition ◽  
New Method ◽  
Subdirectly Irreducible ◽  
Natural Numbers ◽  
Finitely Generated ◽  
Congruence Distributivity ◽  
Finite Height ◽  
Descending Chain Condition ◽  
Simple Algebras

A new method of constructing commutative BCK-algebras is given. It depends upon the notion of a valuation of a lower semilattice in a given commutative BCK-algebra. Any tree vith the descending chain condition has a valuation in the natural numbers, considered as a commutative BCK-algebra; the valuation is the height-function. Thus, any tree of finite height possesses a uniquely determined commutative BCK-structure. The finite trees with at most one atom and height at most n are precisely the finitely generated subdirectly irreducible (simple) algebras in the subvariety of commutative BCK-algebras which satisfy the identity (En): xyn = xyn+1. Due to congruence-distributivity, it is then possible to describe the associated lattice of subvarieties.

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