Browsing by Author "Kinyon, Michael"
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- An elegant 3-basis for inverse semigroupsPublication . Araújo, João; Kinyon, MichaelAbstract It is well known that in every inverse semigroup the binary operation and the unary operation of inversion satisfy the following three identities: x = (xx′)x, (xx′)(y′y) = (y′y)(xx′), (xy)z = x(yz′′). The goal of this note is to prove the converse, that is, we prove that an algebra of type ⟨2, 1⟩ satisfying these three identities is an inverse semigroup and the unary operation coincides with the usual inversion on such semigroups.
- Axioms for unary semigroups via division operationsPublication . Araújo, João; Kinyon, MichaelWhen a semigroup has a unary operation, it is possible to define two bin ary operations, namely, left and right division. In addition it is well known that groups can be defined in terms of those two divisions. The aim of this paper is to extend those results to other classes of unary semigroups. In the first part of the paper we provide characterizations for several classes of unary semigroups, including (a special class of) E-inversive, regular, completely regular, inverse, Clifford, etc., in terms of left and right division. In the second part we solve a problem that was posed elsewhere. The paper closes with a list of open problems.
- A characterization of adequate semigroups by forbidden subsemigroupsPublication . Araújo, João; Kinyon, Michael; Malheiro, AntónioA semigroup is \emph{amiable} if there is exactly one idempotent in each R∗-class and in each L∗-class. A semigroup is \emph{adequate} if it is amiable and if its idempotents commute. We characterize adequate semigroups by showing that they are precisely those amiable semigroups which do not contain isomorphic copies of two particular nonadequate semigroups as subsemigroups.
- Independent axiom systems for nearlatticesPublication . Araújo, João; Kinyon, MichaelA nearlattice is a join semilattice such that every principal filter is a lattice with respect to the induced order. Hickman and later Chajda et al independently showed that nearlattices can be treated as varieties of algebras with a ternary operation satisfying certain axioms. Our main result is that the variety of nearlattices is 2-based, and we exhibit an explicit system of two independent identities. We also show that the original axiom systems of Hickman as well as that of Chajda et al are dependent.
- Inverse semigroups with idempotent-fixing automorphismsPublication . Araújo, João; Kinyon, MichaelA celebrated result of J. Thompson says that if a finite group G has a fixedpoint-free automorphism of prime order, then G is nilpotent. The main purpose of this note is to extend this result to finite inverse semigroups. An earlier related result of B. H. Neumann says that a uniquely 2-divisible group with a fixed-point-free automorphism of order 2 is abelian. We similarly extend this result to uniquely 2-divisible inverse semigroups.
- Matrix theory for independence algebrasPublication . Araújo, João; Bentz, Wolfram; Cameron, Peter; Kinyon, Michael; Konieczny, JanuszA universal algebra A with underlying set A is said to be a matroid algebra if (A, 〈·〉), where 〈·〉 denotes the operator subalgebra generated by, is a matroid. A matroid algebra is said to be an independence algebra if every mapping α : X → A defined on a minimal generating X of A can be extended to an endomorphism of A. These algebras are particularly well-behaved generalizations of vector spaces, and hence they naturally appear in several branches of mathematics, such as model theory, group theory, and semigroup theory. It is well known that matroid algebras have a well-defined notion of dimension. Let A be any independence algebra of finite dimension n, with at least two elements. Denote by End(A) the monoid of endomorphisms of A. In the 1970s, Glazek proposed the problem of extending the matrix theory for vector spaces to a class of universal algebras which included independence algebras. In this paper, we answer that problem by developing a theory of matrices for (almost all) finite-dimensional independence algebras. In the process of solving this, we explain the relation between the classification of inde- pendence algebras obtained by Urbanik in the 1960s, and the classification of finite indepen- dence algebras up to endomorphism-equivalence obtained by Cameron and Szab ́o in 2000. (This answers another question by experts on independence algebras.) We also extend the classification of Cameron and Szab ́o to all independence algebras. The paper closes with a number of questions for experts on matrix theory, groups, semi- groups, universal algebra, set theory or model theory.
- Minimal paths in the commuting graphs of semigroupsPublication . Araújo, João; Kinyon, Michael; Konieczny, JanuszLet S be a finite non-commutative semigroup. The commuting graph of S, denoted G(S), is the graph whose vertices are the non- central elements of S and whose edges are the sets {a, b} of vertices such that a �= b and ab = ba. Denote by T(X) the semigroup of full transformations on a finite set X . Let J be any ideal of T (X ) such that J is different from the ideal of constant transformations on X. We prove that if |X| ≥ 4, then, with a few exceptions, the diameter of G(J ) is 5. On the other hand, we prove that for every positive integer n, there exists a semigroup S such that the diameter of G(S) is n. We also study the left paths in G(S), that is, paths a1 − a2 − ··· − am such that a1 �= am and a1ai = amai for all i ∈ {1,...,m}. We prove that for every positive integer n ≥ 2, except n = 3, there exists a semigroup whose shortest left path has length n. As a corollary, we use the previous results to solve a purely algebraic old problem posed by B.M. Schein.
- On a problem of M. Kambites regarding abundant semigroupsPublication . Araújo, João; Kinyon, MichaelA 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.