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*).

STABLE EQUIVALENCE PROBLEMS FOR FREE ALGEBRAS WITH THE NIELSEN-SCHREIER PROPERTY

2001 ◽  
Vol 11 (06) ◽  
pp. 779-786 ◽  
Author(s):  
ALEXANDER A. MIKHALEV ◽  
JIE-TAI YU
Keyword(s):  
Free Algebra ◽  
Maximal Rank ◽  
Variety Of Algebras ◽  
Free Algebras ◽  
Stable Equivalence ◽  
Two Systems ◽  
Equivalence Problems ◽  
Injective Endomorphism

A variety of algebras is said to be Schreier if any subalgebra of a free algebra of this variety is free in the same variety of algebras. For free algebras of finite ranks of Schreier varieties we prove that if two systems of elements are stably equivalent, then they are equivalent. We define the rank of an endomorphism of a free algebra of a Schreier variety and prove that an injective endomorphism of maximal rank does not change the rank of elements of maximal rank.

scholarly journals Additive primitive length in relatively free algebras

2019 ◽  
Vol 29 (05) ◽  
pp. 849-859
Author(s):  
Vesselin Drensky
Keyword(s):  
Recent Result ◽  
Lie Algebras ◽  
Free Algebra ◽  
Upper Bound ◽  
Polynomial Algebra ◽  
Variety Of Algebras ◽  
Free Algebras ◽  
Primitive Elements ◽  
Number Formula ◽  
Algebra Over A Field

The additive primitive length of an element [Formula: see text] of a relatively free algebra [Formula: see text] in a variety of algebras [Formula: see text] is equal to the minimal number [Formula: see text] such that [Formula: see text] can be presented as a sum of [Formula: see text] primitive elements. We give an upper bound for the additive primitive length of the elements in the [Formula: see text]-generated polynomial algebra over a field of characteristic 0, [Formula: see text]. The bound depends on [Formula: see text] and on the degree of the element. We show that if the field has more than two elements, then the additive primitive length in free [Formula: see text]-generated nilpotent-by-abelian Lie algebras is bounded by 5 for [Formula: see text] and by 6 for [Formula: see text]. If the field has two elements only, then our bounds are 6 for [Formula: see text] and 7 for [Formula: see text]. This generalizes a recent result of Ela Aydın for two-generated free metabelian Lie algebras. In all cases considered in the paper, the presentation of the elements as sums of primitive elements can be found effectively in polynomial time.

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.

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 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 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.]

Wreath products along the period-doubling route to chaos

1999 ◽  
Vol 19 (6) ◽  
pp. 1617-1636 ◽  
Author(s):  
J. D. H. SMITH
Keyword(s):  
Wreath Product ◽  
Period Doubling ◽  
Periodic Points ◽  
Wreath Products ◽  
Explicit Description ◽  
Quadratic Maps ◽  
Combinatorial Description ◽  
Product Construction ◽  
Renormalization Operator ◽  
Route To Chaos

The wreath-product construction is used to give a complete combinatorial description of the dynamics of period-doubling quadratic maps leading to the Feigenbaum map. An explicit description of the action on periodic points uses the Thue–Morse sequence. In particular, a wreath-product construction of this sequence is given. The combinatorial renormalization operator on the period-doubling family of maps is invertible.

scholarly journals Wreath Products of Permutation Classes

10.37236/964 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Robert Brignall
Keyword(s):  
Wreath Product ◽  
Wreath Products ◽  
General Construction ◽  
Finitely Based ◽  
Product Construction

A permutation class which is closed under pattern involvement may be described in terms of its basis. The wreath product construction $X\wr Y$ of two permutation classes $X$ and $Y$ is also closed, and we exhibit a family of classes $Y$ with the property that, for any finitely based class $X$, the wreath product $X\wr Y$ is also finitely based. Additionally, we indicate a general construction for basis elements in the case where $X\wr Y$ is not finitely based.

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