Equational theories in commutative idempotent monoids
It might well be that my question is trivial but I'm not a mathematician,
I just need a formalization for algebraic semiotics.
If I have a commutative idempotent monoid, I can define a partial ordering
$\preccurlyeq$ as follows: $a \preccurlyeq b \equiv_{df} a \cdot b = b$.
Let $E$ be a set of equational axioms (identities) $\{a_1\approx
b_1,\dots,a_n\approx b_n\}$ (over elements $a_i,b_i$ of the monoid). I
define $\doteq_E$ as the least relation of which $E\subseteq\doteq_E$,
reflexivity, assymetry and transitivity hold and further $a\doteq_E b$
implies $a\cdot c \doteq_E b\cdot c$ for all $a,b,c$. Is this relation a
congruence?
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