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}.

scholarly journals Free Algebras Over a Poset in Varieties of Łukasiewicz–Moisil Algebras

2015 ◽  
Vol 48 (3) ◽  
Author(s):  
A. Figallo Orellano ◽  
C. Gallardo
Keyword(s):  
Free Algebra ◽  
General Construction ◽  
Free Algebras ◽  
Finitely Generated ◽  
Finite Poset

AbstractA general construction of the free algebra over a poset in varieties finitely generated is given in [8]. In this paper, we apply this to the varieties of Łukasiewicz-Moisil algebras, giving a detailed description of the free algebra over a finite poset (X, ≤) , Free

Finite Projective Distributive Lattices

1970 ◽  
Vol 13 (1) ◽  
pp. 139-140 ◽  
Author(s):  
G. Grätzer ◽  
B. Wolk
Keyword(s):  
Distributive Lattice ◽  
Free Algebra ◽  
Distributive Lattices ◽  
Free Algebras ◽  
Algebra Ii

The theorem stated below is due to R. Balbes. The present proof is direct; it uses only the following two well-known facts: (i) Let K be a category of algebras, and let free algebras exist in K; then an algebra is projective if and only if it is a retract of a free algebra, (ii) Let F be a free distributive lattice with basis {xi | i ∊ I}; then ∧(xi | i ∊ J0) ≤ ∨(xi | i ∊ J1) implies J0∩J1≠ϕ. Note that (ii) implies (iii): If for J0 ⊆ I, a, b ∊ F, ∧(xi | i ∊ J0)≤a ∨ b, then ∧ (xi | i ∊ J0)≤ a or b.

scholarly journals Bounded endomorphisms of free P-algebras

1992 ◽  
Vol 34 (2) ◽  
pp. 209-214
Author(s):  
Daniel Ševčovič
Keyword(s):  
Free Algebra ◽  
Present Note ◽  
Finitely Generated ◽  
Efficient Tool ◽  
Pseudocomplemented Lattices

The present note deals with bounded endomorphisms of free p-algebras (pseudocomplemented lattices). The idea of bounded homomorphisms was introduced by R. McKenzie in [8]. T. Katriňák [5] subsequently studied the properties of bounded homomorphisms for the varieties of p-algebras. This concept is also an efficient tool for the characterization of, so-called, splitting as well as projective algebras in the varieties of all lattices or p-algebras. For details the reader is referred to [2], [5], [6], [7] and other references therein. Let us emphasize that the main results that are contained in the above mentioned references strongly depend on the boundedness of each endomorphism of any finitely generated free algebra in a given variety.

scholarly journals THE AUTOMORPHISM GROUP OF A SHIFT OF LINEAR GROWTH: BEYOND TRANSITIVITY

2015 ◽  
Vol 3 ◽  
Author(s):  
VAN CYR ◽  
BRYNA KRA
Keyword(s):  
Automorphism Group ◽  
Finite Groups ◽  
Linear Growth ◽  
Finite Alphabet ◽  
Finitely Generated ◽  
Complexity Function ◽  
Algebraic Properties

For a finite alphabet ${\mathcal{A}}$ and shift $X\subseteq {\mathcal{A}}^{\mathbb{Z}}$ whose factor complexity function grows at most linearly, we study the algebraic properties of the automorphism group $\text{Aut}(X)$. For such systems, we show that every finitely generated subgroup of $\text{Aut}(X)$ is virtually $\mathbb{Z}^{d}$, in contrast to the behavior when the complexity function grows more quickly. With additional dynamical assumptions we show more: if $X$ is transitive, then $\text{Aut}(X)$ is virtually $\mathbb{Z}$; if $X$ has dense aperiodic points, then $\text{Aut}(X)$ is virtually $\mathbb{Z}^{d}$. We also classify all finite groups that arise as the automorphism group of a shift.

scholarly journals Ideal Dasar Prima Dalam Aljabar Atas Suatu Ring Komutatif

2018 ◽  
Vol 7 (2) ◽  
pp. 79-86
Author(s):  
Khurul Wardati
Keyword(s):  
Commutative Ring ◽  
Free Algebra ◽  
Free Algebras

Definisi ideal dasar dan ideal bebas dalam aljabar bebas atas ring komutatif dengan elemen satuan adalah ekuivalen. Namun, ideal dasar dalam suatu aljabar tak bebas belum tentu merupakan ideal bebas, sementara ideal bebas pasti ideal dasar. Artikel ini membahas beberapa sifat ideal dasar prima dalam aljabar tak bebas atas ring komutatif dengan elemen satuan. [The definitions of basic ideal and free ideal in free algebras over a unital commutative ring are equivalen. However, a basic ideal in the non-free algebra is not neceearily a free ideal, while any free ideal is definitely a basic ideal. This paper will discuss some properties of prime basic ideal in non-free algebras over a unital commutative ring.]

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.

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 Bellissima’s construction of the finitely generated free Heyting algebras, and beyond

2010 ◽  
Vol 49 (7-8) ◽  
pp. 743-771 ◽  
Author(s):  
Luck Darnière ◽  
Markus Junker
Keyword(s):  
Heyting Algebras ◽  
Finitely Generated

scholarly journals The persistence of universal formulae in free algebras

1987 ◽  
Vol 36 (1) ◽  
pp. 11-17 ◽  
Author(s):  
Anthony M. Gaglione ◽  
Dennis Spellman
Keyword(s):  
Wreath Product ◽  
Free Algebra ◽  
Variety Of Algebras ◽  
Free Algebras ◽  
Product Construction

Gilbert Baumslag, B.H. Neumann, Hanna Neumann, and Peter M. Neumann successfully exploited their concept of discrimination to obtain generating groups of product varieties via the wreath product construction. We have discovered this same underlying concept in a somewhat different context. Specifically, let V be a non-trivial variety of algebras. For each cardinal α let Fα(V) be a V-free algebra of rank α. Then for a fixed cardinal r one has the equivalence of the following two statements:(1) Fr(V) discriminates V. (1*) The Fs(V) satisfy the same universal sentences for all s≥r. Moreover, we have introduced the concept of strong discrimination in such a way that for a fixed finite cardinal r the following two statements are equivalent:(2) Fr(V) strongly discriminates V. (2*) The Fs(V) satisfy the same universal formulas for all s ≥ r whenever elements of Fr(V) are substituted for the unquantified variables. On the surface (2) and (2*) appear to be stronger conditions than (1) and (1*). However, we have shown that for particular varieties (of groups) (2) and (2*) are no stronger than (1) and (1*).

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