Relatively congruence modular quasivarieties of modules

We show that the quasiequational theory of a relatively congruence modular quasivariety of left $R$-modules is determined by a two-sided ideal in $R$ together with a filter of left ideals. The two-sided ideal encodes the identities that hold in the quasivariety, while the filter of left ideals encodes the quasiidentities. The filter of left ideals defines a generalized notion of torsion. It follows from our result that if $R$ is left Artinian, then any relatively congruence modular quasivariety of left $R$-modules is axiomatizable by a set of identities together with at most one proper quasiidentity, and if $R$ is a commutative Artinian ring then any relatively congruence modular quasivariety of left $R$-modules is a variety.Comment: 11 page

Dualizable algebras with parallelogram terms

We prove that if A is a finite algebra with a parallelogram term that satisfies the split centralizer condition, then A is dualizable. This yields yet another proof of the dualizability of any finite algebra with a near unanimity term, but more importantly proves that every finite module, group or ring in a residually small variety is dualizable

Self-rectangulating varieties of type 5

We show that a locally finite variety which omits abelian types is self-regulating if and only if it has a compatible semilattice term operation. Such varieties must have a type-set {5}. These varieties are residually small and, when they are finitely generated, they have definable principal congruences. We show that idempotent varieties with a compatible semilattice term operation have the congruence extension property

The relationship between two commutators

We clarify the relationship between the linear commutator and the ordinary commutator by showing that in any variety satisfying a nontrivial idempotent Mal'cev condition the linear commutator is definable in terms of the centralizer relation. We derive from this that abelian algebras are quasi-affine in such varieties. We refine this by showing that if A is an abelian algebra and V(A) satifies an idempotent Mal'cev condition which fails to hold in the variety of semilattices, then A is affine

A note on "A minimal congruence lattice representation for Mp+1\mathbb M_{p+1}''

We reprove a theorem of Bunn, Grow, Insall, and Thiem, which asserts that a minimal congruence lattice representation for $\mathbb M_{p+1}$ has size $2p$, and is an expansion of a regular $D_{2p}$-set.Comment: 2 page

Varieties whose finitely generated members are free

We prove that a variety of algebras whose finitely generated members are free must be definitionally equivalent to the variety of sets, the variety of pointed sets, a variety of vector spaces over a division ring, or a variety of affine vector spaces over a division ring.Comment: 17 page

Divisibility Theory of Commutative Rings and Ideal Distributivity

We begin by investigating the class of commutative unital rings in which no two distinct elements divide the same elements. We prove that this class forms a finitely axiomatizable, relatively ideal distributive quasivariety, and it equals the quasivariety generated by the class of integral domains with trivial unit group. We end the paper by proving a representation theorem that provides more evidence to the conjecture that B\'ezout monoids describe exactly the monoids of finitely generated ideals of commutative unital rings with distributive ideal lattice

Growth Rates of Algebras, II: Wiegold Dichotomy

We investigate the function $d_\mathbf{A}(n)$, which gives the size of a least size generating set for $\mathbf{A}^n$, in the case where $\mathbf{A}$ has a cube term. We show that if $\mathbf{A}$ has a $k$-cube term and $\mathbf{A}^k$ is finitely generated, then $d_\mathbf{A}(n) \in O(\log(n))$ if $\mathbf{A}$ is perfect and $d_\mathbf{A}(n) \in O(n)$ if $\mathbf{A}$ is imperfect. When $\mathbf{A}$ is finite, then one may replace "Big Oh" with "Big Theta" in these estimates.Comment: Second paper in a series of three, but complete in itsel

Growth Rates of Algebras, III: Growth Rates of Solvable Algebras

We investigate how the behavior of the function d_A(n) that gives the size of a least size generating set for A^n, influences the structure of a finite solvable algebra A.Comment: 25 pages. Third paper in a series, but independently readabl

Is supernilpotence super nilpotence?

We show that the answer to the question in the title is: ``Yes, for finite algebras.''Comment: 11 pages. Key words and phrases: Higher commutator, congruence, nilpotent, supernilpotent, tame congruence theory, twin monoi