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Matemática e Estatística / Mathematics and Statistics

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Browsing Matemática e Estatística / Mathematics and Statistics by Subject "20M20"

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  • On a problem of M. Kambites regarding abundant semigroups
    Publication . Araújo, João; Kinyon, Michael
    A semigroup is regular if it contains at least one idempotent in each -class and in each L-class. A regular semigroup is inverse if it satisfies either of the following equivalent conditions: (i) there is a unique idempotent in each -class and in each L-class, or (ii) the idempotents commute. Analogously, a semigroup is abundant if it contains at least one idempotent in each *-class and in each L*-class. An abundant semigroup is adequate if its idempotents commute. In adequate semigroups, there is a unique idempotent in each * and L*-class. M. Kambites raised the question of the converse: in a finite abundant semigroup such that there is a unique idempotent in each * and L*-class, must the idempotents commute? In this note, we provide a negative answer to this question.
    2012Journal article Open access Show more
  • Semigroups of transformations preserving an equivalence relation and a cross-section
    Publication . Araújo, João; Konieczny, Janusz
    For a set X, an equivalence relation ρ on X, and a cross-section R of the partition X/ρ induced by ρ, consider the semigroup T (X, ρ,R) consisting of all mappings a from X to X such that a preserves both ρ (if (x, y) ∈ ρ then (xa, ya) ∈ ρ) and R (if r ∈ R then ra ∈ R). The semigroup T (X, ρ,R) is the centralizer of the idempotent transformation with kernel ρ and image R. We determine the structure of T (X, ρ,R) in terms of Green’s relations, describe the regular elements of T (X, ρ,R), and determine the following classes of the semigroups T (X, ρ,R): regular, abundant, inverse, and completely regular.
    2004Journal article Open access Show more