Univalent Semigroud

A univalent semigroud is an Idempotent Semigroud where all elements are Bicommutative; EG:

$[a,a,[b,b,x]]=[b,b,[a,a,x]]$
$[[x,a,a],b,b]=[[x,b,b],a,a]$

Somewhat shockingly, this means that the relation $a\leq b=[a,b,a]=a$ is Reflexive and Antisymmetric, which means that $a\leq b\land b\leq a$ forms an Identity System!

The proof is quite elementary. Reflexivity follows directly from idempotency. Antisymmetry is a bit trickier. First, a lemma, for every $a,b$ , if $a\leq b$ , then $a=[a,b,b]$

$\displaystyle a$	$\displaystyle=[a,b,a]$	$\displaystyle(a\leq b)$
	$\displaystyle=[a,a,a,b,b,b,a]$	$\displaystyle(\text{idempotency})$
	$\displaystyle=[a,b,b,a,a,b,a]$	$\displaystyle(\text{bicommutativity})$
	$\displaystyle=[a,b,b,a,a]$	$\displaystyle(a\leq b)$
	$\displaystyle=[a,a,a,b,b]$	$\displaystyle(\text{bicommutativity})$
	$\displaystyle=[a,b,b]$	$\displaystyle(\text{idempotency})$

A similar line of reasoning lets us deduce that $a\leq b$ implies $a=[b,b,a]$ .

Now, suppose that $a\leq b\land b\leq a$ . We get

$\displaystyle a$	$\displaystyle=[a,b,b]$	$\displaystyle(\text{lemma})$
	$\displaystyle=[a,[a,a,b],b]$	$\displaystyle(\text{lemma})$
	$\displaystyle=[[a,a,a],b,b]$	$\displaystyle(\text{para-associativity})$
	$\displaystyle=[[a,b,b],a,a]$	$\displaystyle(\text{bicommutativity})$
	$\displaystyle=[b,a,a]$	$\displaystyle(\text{lemma})$
	$\displaystyle=b$	$\displaystyle(\text{lemma})$