I define a kind of template or macro to be used in a slightly odd fashion,
not exactly typed second order logic. Let Ab(t,A,B) be the macro
that expands (when appropriate inputs are given) to the logical expression
(t(A,xbar) = t(A,ybar)) implies (t(B,xbar) = t(B,ybar)) . Similarly for SAb(t,B,C,D),
using (t(C,xbar)=t(D,ybar)) implies (t(B,xbar)=t(B,ybar)). Now for a given algebra
AA, ((for any xbar and ybar which are tuples of appropriate length built from
(the underlying set of) AA, for any term t from (the set of term operations of) AA,
[For any a,b from AA Ab(t,a,b) holds] )) iff AA is abelian.
Replacing the "last line" in the above with
[For any b,c,d from AA Sab(t,b,c,d) holds] )) iff AA is strongly abelian
gets the definition of a strongly abelian algebra as well.
There are variants of the above defintions where binary terms t(a,x) instead of
larger arity terms t(a,xbar) are used, and for two congruences $\alpha \leq \beta$
of AA one can define a generalization ($\beta$ is abelian or strongly abelian over
$\alpha$) using $\alpha$-related in place of = and
asking for certain of a,b,c,d and the bars to be $\beta$-related. Also, once the
defintion is understood for an algebra AA, it can be extended to apply to classes
A web search for strongly abelian universal algebra leads to various papers in
the literature, with Kiss, McKenzie, and Valeriote among the authors. Kiss and
Valeriote in a paper on Abelian Algebras and the Hamiltonian Property mention
some of the literature and note that matrix powers of unary algebras provide
basic examples of strongly Abelian algebras. In another paper on strongly abelian
varieties these same two authors mention the result that strongly abelian algebras
have finite essential arity, and that something similar holds for locally finite strongly
I know of no nice way of expressing essential arity in terms of identities. The not so
nice way involves picking an integer n and then a certain set of formulas which are
tantamount to saying t(abar,xbar)=t(abar,ybar), except one needs to single out the
inessential variables where they live rather than conveniently grouping them together
into xbar or ybar.
I have no specific example of an algebra that is not strongly abelian, but is of bounded
essential arity. Here is an idea on how to build one though a finite such algebra. Take
a universe of size n at least 4. (Smaller might work, but I want enough room for success.)
Order the set as a chain, with 0 as the least. Create an operation of desired arity and call it b
and make sure it is not strongly abelian by ensuring that b(a,xbar) is different from b(a,ybar)
for at least one valuation of a, xbar,and ybar, while making it agree for c,xbar and d,ybar.
Compatible with that condition, let b have the value v satisfy that it is smaller than any
of the values of its arguments, unless one of them is 0 in which case 0=b(a,ybar). Now any
term which has a depth of n many b's will evaluate to 0, and be constant. If you take
sufficient care, you can show that this algebra has essential arity at most w^n for w the essential arity of b.
Gerhard "Ask Me About System Design" Paseman, 2013.04.24