Subalgebras of the squares of weakly diagonal majority algebras

If A is a minimal algebra (that is, has no proper subalgebras) then the set S2(A) of all subalgebras of A2 has a natural structure of ordered involutive monoid. This paper gives a characterization of monoids S that appear in the role of this monoid if A is finite, weakly diagonal (every subalgebra of A2 contains the graph of an automorphism of A) and has a majority term.

Minimal Characteristic Algebras for Leftmost k-Normal Identities

A characteristic algebra for a hereditary property of identities is an algebra [Formula: see text] which generates the variety determined by all identities with that property. We use Płonka's construction, and known minimal characteristic algebras for the k-normality and leftmost properties, to construct minimal characteristic algebras of type (2) for leftmost k-normality for 1 ≤ k ≤ 3, and show that Płonka's construction does not always give a minimal algebra.

On Minimal Norms onMn

We show that for each minimal normN(⋅)on the algebraℳnof alln×ncomplex matrices, there exist norms‖⋅‖1and‖⋅‖2onℂnsuch thatN(A)=max{‖Ax‖2:‖x‖1=1, x∈ℂn}for allA∈ℳn. This may be regarded as an extension of a known result on characterization of minimal algebra norms.

RESIDUAL SMALLNESS AND WEAK CENTRALITY

We develop a method of creating skew congruences on subpowers of finite algebras using groups of twin polynomials, and apply it to the investigation of residually small varieties generated by nilpotent algebras. We prove that a residually small variety generated by a finite nilpotent (in particular, a solvable E-minimal) algebra is weakly abelian. Conversely, we show in two special cases that a weakly abelian variety is residually bounded by a finite number: when it is generated by an E-minimal, or by a finite strongly nilpotent algebra. This establishes the RS-conjecture for E-minimal algebras.

AN ORDER-THEORETIC PROPERTY OF THE COMMUTATOR

We describe a new order-theoretic property of the commutator for finite algebras. As a corollary we show that any right nilpotent congruence on a finite algebra is left nilpotent. The result is false for infinite algebras and the converse is false even for finite algebras. We show further that any solvable E-minimal algebra is left nilpotent, any finite algebra whose congruence lattice contains a 0, 1-sublattice isomorphic to M3 is left nilpotent and any homomorphic image of a finite abelian algebra is left and right nilpotent.