8 documents found, page 1 of 1
Matrix theory for independence algebras
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 the...
Inverse semigroups with idempotent-fixing automorphisms
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-divi...
A characterization of adequate semigroups by forbidden subsemigroups
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.
On a problem of M. Kambites regarding abundant semigroups
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 abu...
Axioms for unary semigroups via division operations
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 sp...
An elegant 3-basis for inverse semigroups
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 ...
Independent axiom systems for nearlattices
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 als...
Minimal paths in the commuting graphs of semigroups
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 tha...