International Journal of Algebra and Computation

We describe an algorithm for explicitly listing the irreducible monomial subgroups of GL(n,q), given a suitable list of finite irreducible monomial subgroups of , where n is 4 or a prime, and q is a prime power. Particular attention is paid to the case n=4, and the algorithm is illustrated for n=4 and q=5. Certain primitive permutation groups can be constructed from a list of irreducible monomial subgroups of GL(n,q). The paper's final section shows that the computation of automorphisms of such permutation groups reduces mainly to computation of irreducible monomial subgroups of GL(n,q), q prime.

An involuted semilattice <S,∨,^-> is a semilattice <S,∨> with an involution ^-: S→S, i.e., <S,∨,^-> satisfies , and . In this paper we study the properties of such semilattices. In particular, we characterize free involuted semilattices in terms of ordered pairs of subsets of a set. An involuted semilattice <S,∨,^-,1> with greatest element 1 is said to be complemented if it satisfies a∨ā=1. We also characterize free complemented semilattices. We next show that complemented semilattices are related to ternary algebras. A ternary algebra <T,+,*,^-,0,ϕ,1> is a de Morgan algebra with a third constant ϕ satisfying , and (a+ā)+ϕ=a+ā. If we define a third binary operation ∨ on T as a∨b=a*b+(a+b)*ϕ, then <T,∨,^-,ϕ> is a complemented semilattice.

For positive integers n and c, with n≥2, let G_n,c be a relatively free group of rank n in the variety N₂A∧AN₂∧N_c. It is shown that there exists an explicitly described finite subset Ω of IA-automorphisms of G_n,c such that the cardinality of Ω is independent upon n and c and the subgroup of the automorphism group Aut(G_n,c) of G_n,c generated by the tame automorphisms and Ω has finite index in Aut(G_n,c). This is a simpler result than one given in [12, Theorem 1(I)]. Let L(G_n,c) be the associated Lie ring of G_n,c and K be a field of characteristic zero. The method developed in the proof of the aforementioned result is applied in order to find an explicitly described finite subset Ω_L of the IA-automorphism group of K⊗L(G_n,c) such that the automorphism group of K⊗L(G_n,c) is generated by GL(n,K) and Ω_L. In particular, for n≥3, the cardinality of Ω_L is independent upon n and c.

Using techniques pioneered by R. McKenzie, we prove that there is no algorithm which, given a finite algebra in a finite language, determines whether the variety (equational class) generated by the algebra has a model companion. In particular, there exists a finite algebra such that the variety it generates has no model companion; this answers a question of Burris and Werner from 1979.

For a partially ordered set P, let Co(P) denote the lattice of all order-convex subsets of P. For a positive integer n, we denote by (resp., SUB(n)) the class of all lattices that can be embedded into a lattice of the form

(1) Both classes and SUB(n), for any positive integer n, are locally finite, finitely based varieties of lattices, and we find finite equational bases of these varieties.

(2) The variety is the quasivariety join of all the varieties SUB(n), for 1≤n<ω, and it has only countably many subvarieties. We classify these varieties, together with all the finite subdirectly irreducible members of .

(3) Every finite subdirectly irreducible member of is projective within , and every subquasivariety of is a variety.

Let L be a finitely generated free Lie algebra. We construct an example of two elements u and v of L such that the factor algebras L/(u) and L/(v) are isomorphic, where (u) and (v) are ideals of L generated by u and v, respectively, but there is no automorphism φ of L such that φ(u)=v.