Commutator theory

Remarks

The least such $\delta$ exists: $C(\alpha, \beta; 1_A)$ holds for the full congruence $1_A$, and if $C(\alpha, \beta; \delta_i)$ holds for every $i$, then $C(\alpha, \beta; \bigcap_i \delta_i)$ holds. The definition is often called the term condition, and this commutator the term-condition commutator. Example: in a group $G$, let $\alpha$ and $\beta$ be the congruences of normal subgroups $M$ and $N$. Then $[\alpha, \beta]$ is the congruence of the commutator subgroup $[M, N]$. In particular $[1_G, 1_G]$ is the congruence of the commutator subgroup of $G$. Non-example of $C(1_G, 1_G; 0_G)$: let $G$ be a nonabelian group with $b c \ne c b$, and let $t(x, y) = x y x^{-1} y^{-1}$. With $a = e$, $\bar{c} = c$ and $\bar{d} = e$, both $t(e, c)$ and $t(e, e)$ equal $e$, but $t(b, c) = b c b^{-1} c^{-1} \ne e = t(b, e)$. In a congruence-modular variety the commutator is symmetric, $[\alpha, \beta] = [\beta, \alpha]$, and $[\alpha, \beta] \le \alpha \wedge \beta$.