Abelian algebra

Remarks

In the language of commutator theory, $A$ is abelian if and only if $[1_A, 1_A] = 0_A$, where $1_A$ and $0_A$ are the largest and the least congruences of $A$. Example: every module $M$ over a ring $R$ is abelian. Each term operation has the form $t(x, \bar{y}) = r x + s_1 y_1 + \cdots + s_n y_n$ with $r, s_i \in R$. So $t(a, \bar{c}) = t(a, \bar{d})$ means $\sum_i s_i c_i = \sum_i s_i d_i$, and this gives $t(b, \bar{c}) = t(b, \bar{d})$ for every $b$. Non-example: a lattice $L$ with elements $0 < 1$ is not abelian. Take $t(x, y) = x \wedge y$, $a = 0$, $c = 0$ and $d = 1$: then $t(0, 0) = 0 = t(0, 1)$, but $t(1, 0) = 0 \ne 1 = t(1, 1)$. A group is abelian in this sense exactly when it is commutative.