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For [Formula: see text] a positive integer and [Formula: see text] a finite set of finite algebras, let [Formula: see text] denote the largest [Formula: see text]-generated subdirect product whose subdirect factors are algebras in [Formula: see text]. When [Formula: see text] is the set of all [Formula: see text]-generated subdirectly irreducible algebras in a locally finite variety [Formula: see text], then [Formula: see text] is the free algebra [Formula: see text] on [Formula: see text] free generators for [Formula: see text]. For a finite algebra [Formula: see text] the algebra [Formula: see text] is the largest [Formula: see text]-generated subdirect power of [Formula: see text]. For every [Formula: see text] and finite [Formula: see text] we provide an upper bound on the cardinality of [Formula: see text]. This upper bound depends only on [Formula: see text] and these basic parameters: the cardinality of the automorphism group of [Formula: see text], the cardinalities of the subalgebras of [Formula: see text], and the cardinalities of the equivalence classes of certain equivalence relations arising from congruence relations of [Formula: see text]. Using this upper bound on [Formula: see text]-generated subdirect powers of [Formula: see text], as [Formula: see text] ranges over the [Formula: see text]-generated subdirectly irreducible algebras in [Formula: see text], we obtain an upper bound on [Formula: see text]. And if all the [Formula: see text]-generated subdirectly irreducible algebras in [Formula: see text] have congruence lattices that are chains, then we characterize in several ways those [Formula: see text] for which this upper bound is obtained.

Weak Stone-Ockham algebras are those algebras (L; ∧, ∨, f, ⋆,0,1) of type 〈2,2,1,1,0,0〉, where (L; ∧, ∨, f,0,1) is an Ockham algebra, (L; ∧, ∨, ⋆,0,1) is a weak Stone algebra, and the unary operations f and ⋆ commute. In this paper, we give a complete description of the structure of the lattice of congruences on the subdirectly irreducible algebras.

Let bdpO be the variety consisting of those algebras (L; ∧, ∨, f, *,0,1) of type 〈2,2,1,1,0,0〉, where (L; ∧, ∨, f,0,1) is an Ockham algebra, (L; ∧, ∨, *,0,1) is a demi-pseudocompleted lattice, and the unary operations f and * satisfy the conditions [f(x)]*=f2(x) and f(x*)=x**. Here we give a description of the structure of the subdirectly irreducible algebras in bdpO.

For a wide class of double Ockham algebras we characterise the subdirectly irreducible members by determining the structure of their lattice of congruences.

Let [Formula: see text] be a variety of the form [Formula: see text] where [Formula: see text] is a finite subdirectly irreducible algebra. We show that if [Formula: see text] is naturally dualizable (in the sense of D. M. Clark and B. A. Davey, i.e. with respect to the discrete topology) then the variety [Formula: see text][Formula: see text] determined by all normal identities of [Formula: see text] (the so called nilpotent shift of [Formula: see text]) is also naturally dualizable. We give a finite algebra [Formula: see text] and a relational system [Formula: see text], constructed explicitly from the system [Formula: see text] for [Formula: see text], such that [Formula: see text] and [Formula: see text] dualizes [Formula: see text] .

Supplementary finite-dimensional irreducible representations of the dynamical algebra subtending the two-body Calogero model are constructed by pointing out the corresponding differential realizations.

Chapter 12 of "The Structure of Finite Algebras" by D. Hobby and R. McKenzie contains theorems revealing how the set of types appearing in a locally finite variety [Formula: see text] influences the size of the free algebra in [Formula: see text] freely generated by n elements. We provide more results in this vein. If A is a subdirectly irreducible algebra of size k, then a lower bound on the number of n-ary polynomials of A is obtained for each case that the monolith of A has type 3, 4, or 5. Examples for every k show that in each case the lower bound is the best possible. As an application of these results we show that for every finite k if all k-element simple algebras are partitioned into five classes according to their type, then algebras in each class have a sharply determined band of possible values for their free spectra. These five bands are disjoint except for some overlap on simple algebras of types 2 and 5.

We investigate de Morgan algebras with double pseudocomplementation. In particular, we characterize the congruence relations of such algebras, and provide a complete description of subdirectly irreducible members of this class.

An ideal I of a (non-associative) algebra A is dense if the multiplication algebra of A acts faithfully on I, and is complementedly dense if it is a direct summand of a dense ideal. We prove that every complementedly dense ideal of a semiprime algebra is a semiprime algebra, and determine its central closure and its extended centroid. We also prove that a semiprime algebra is an essential subdirect product of prime algebras if and only if, its extended centroid is a direct product of fields. This result is applied to discuss decomposable algebras with respect to some familiar closures for ideals.

We study a variety of modes (idempotent algebras with mutually commuting term operations), so called differential modes, having a strongly solvable chain 0 ≤ α ≤ 1 in their congruence lattices. We show an explicit description of subdirectly irreducible algebras in this variety, and use it to compute residual bounds of its subvarieties. It follows from our results that all subvarieties with a finite residual bound are finitely based.