scholarly journals GLOBALS OF PSEUDOVARIETIES OF COMMUTATIVE SEMIGROUPS: THE FINITE BASIS PROBLEM, DECIDABILITY AND GAPS

2001 ◽  
Vol 44 (1) ◽  
pp. 27-47 ◽  
Author(s):  
Jorge Almeida ◽  
Assis Azevedo
Keyword(s):  
Finite Basis ◽  
Subject Classification ◽  
Finitely Based ◽  
Commutative Semigroups ◽  
Finite Basis Problem ◽  
Basis Problem

AbstractWhereas pseudovarieties of commutative semigroups are known to be finitely based, the globals of monoidal pseudovarieties of commutative semigroups are shown to be finitely based (or of finite vertex rank) if and only if the index is 0, 1 or $\omega$. Nevertheless, on these pseudovarieties, the operation of taking the global preserves decidability. Furthermore, the gaps between many of these globals are shown to be big in the sense that they contain chains which are order isomorphic to the reals.AMS 2000 Mathematics subject classification: Primary 20M07; 20M05

Varieties generated by 2-testable monoids

2012 ◽  
Vol 49 (3) ◽  
pp. 366-389 ◽  
Author(s):  
Edmond Lee
Keyword(s):  
Present Article ◽  
Chain Condition ◽  
Finite Basis ◽  
Finitely Based ◽  
Finitely Generated ◽  
Finite Basis Problem ◽  
Basis Problem ◽  
Descending Chain Condition ◽  
Ascending Chain Condition

The smallest monoid containing a 2-testable semigroup is defined to be a 2-testable monoid. The well-known Brandt monoid B21 of order six is an example of a 2-testable monoid. The finite basis problem for 2-testable monoids was recently addressed and solved. The present article continues with the investigation by describing all monoid varieties generated by 2-testable monoids. It is shown that there are 28 such varieties, all of which are finitely generated and precisely 19 of which are finitely based. As a comparison, the sub-variety lattice of the monoid variety generated by the monoid B21 is examined. This lattice has infinite width, satisfies neither the ascending chain condition nor the descending chain condition, and contains non-finitely generated varieties.

scholarly journals Finite basis problem for semigroups of order six

2015 ◽  
Vol 18 (1) ◽  
pp. 1-129 ◽  
Author(s):  
Edmond W. H. Lee ◽  
Wen Ting Zhang
Keyword(s):  
Present Article ◽  
Basis Property ◽  
Finite Basis ◽  
The Other ◽  
Finitely Based ◽  
Minimal Order ◽  
Finite Basis Problem ◽  
Basis Problem ◽  
Finite Basis Property

AbstractTwo semigroups are distinct if they are neither isomorphic nor anti-isomorphic. Although there exist $15\,973$ pairwise distinct semigroups of order six, only four are known to be non-finitely based. In the present article, the finite basis property of the other $15\,969$ distinct semigroups of order six is verified. Since all semigroups of order five or less are finitely based, the four known non-finitely based semigroups of order six are the only examples of minimal order.

THE VARIETY GENERATED BY A CERTAIN TRANSFORMATION MONOID

2008 ◽  
Vol 18 (07) ◽  
pp. 1193-1201 ◽  
Author(s):  
WENTING ZHANG ◽  
YANFENG LUO
Keyword(s):  
Finite Basis ◽  
Affirmative Answer ◽  
Finitely Based ◽  
Finite Basis Problem ◽  
Basis Problem ◽  
Transformation Monoid

It is shown that the transformation monoid [Formula: see text] is finitely based and a finite identity basis for [Formula: see text] is given, which solves the finite basis problem for [Formula: see text]. It is also shown that the variety Var[Formula: see text] generated by [Formula: see text] for which xyx is not an isoterm has uncountably many subvarieties, which gives an affirmative answer to a question of Jackson.

scholarly journals THE VARIETY GENERATED BY AN AI-SEMIRING OF ORDER THREE

2020 ◽  
Vol 6 (2) ◽  
pp. 117
Author(s):  
Xianzhong Zhao ◽  
Miaomiao Ren ◽  
Siniša Crvenković ◽  
Yong Shao ◽  
Petar Dapić
Keyword(s):  
Finite Basis ◽  
Equational Logic ◽  
Finitely Based ◽  
Finite Basis Problem ◽  
Basis Problem

Up to isomorphism, there are 61 ai-semirings of order three. The finite basis problem for these semirings is investigated. This problem for 45 semirings of them is answered by some results in the literature. The remaining semirings are studied using equational logic. It is shown that with the possible exception of the semiring \(S_7\), all ai-semirings of order three are finitely based.

On the finite basis problem for certain 2-limited words

2012 ◽  
Vol 29 (3) ◽  
pp. 571-590 ◽  
Author(s):  
Jian Rong Li ◽  
Wen Ting Zhang ◽  
Yan Feng Luo
Keyword(s):  
Finite Basis ◽  
Finite Basis Problem ◽  
Basis Problem

On the finite basis problem for the monoids of partial extensive injective transformations

2015 ◽  
Vol 91 (2) ◽  
pp. 524-537
Author(s):  
Xun Hu ◽  
Yuzhu Chen ◽  
Yanfeng Luo
Keyword(s):  
Finite Basis ◽  
Finite Basis Problem ◽  
Basis Problem

scholarly journals The finite basis problem for Kauffman monoids

2015 ◽  
Vol 74 (3-4) ◽  
pp. 333-350 ◽  
Author(s):  
K. Auinger ◽  
Yuzhu Chen ◽  
Xun Hu ◽  
Yanfeng Luo ◽  
M. V. Volkov
Keyword(s):  
Finite Basis ◽  
Finite Basis Problem ◽  
Basis Problem

REFLEXIVE RELATIONS, EXTENSIVE TRANSFORMATIONS AND PIECEWISE TESTABLE LANGUAGES OF A GIVEN HEIGHT

2004 ◽  
Vol 14 (05n06) ◽  
pp. 817-827 ◽  
Author(s):  
M. V. VOLKOV
Keyword(s):  
Complete Solution ◽  
Finite Basis ◽  
Finite Basis Problem ◽  
Basis Problem ◽  
Transformation Monoid ◽  
Single Relation

Straubing deduced from Simon's theorem that monoids of reflexive relations as well as monoids of order preserving extensive transformations generate the pseudovariety of [Formula: see text]-trivial monoids. We refine these results by showing that each pseudovariety in Simon's hierarchy of [Formula: see text]-trivial monoids is generated by a single relation/transformation monoid. From this and from some results by Blanchet–Sadri we obtain a complete solution of the finite basis problem for Straubing's monoids.

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