Let $G$ be a nonempty set together with three binary operations: $\,\sharp\,$ , $\stackrel{{\scriptstyle\rightharpoonup}}{{\cdot}}$ and $\stackrel{{\scriptstyle\leftharpoonup}}{{\cdot}}$ . The set $G$ is called a trisemigroup if the following two properties hold:

(i)

For each $\ast\in\{\,\,\sharp\,,\,\stackrel{{\scriptstyle\rightharpoonup}}{{\cdot}},\,\stackrel{{\scriptstyle\leftharpoonup}}{{\cdot}}\,\}$ , $(G,\,\ast\,)$ is a semigroup.

(ii)

The three binary operations $\,\sharp\,$ , $\stackrel{{\scriptstyle\rightharpoonup}}{{\cdot}}$ and $\stackrel{{\scriptstyle\leftharpoonup}}{{\cdot}}$ satisfy the Hu-Liu triassociative law :

$$(x\stackrel{{\scriptstyle\rightharpoonup}}{{\cdot}}y)\,\sharp\,z=x\,\sharp\,(y\stackrel{{\scriptstyle\leftharpoonup}}{{\cdot}}z)$$

(1)

$$(x\,\sharp\,y)\stackrel{{\scriptstyle\rightharpoonup}}{{\cdot}}z=x\,\sharp\,(y\stackrel{{\scriptstyle\rightharpoonup}}{{\cdot}}z)$$

(2)

$$x\stackrel{{\scriptstyle\leftharpoonup}}{{\cdot}}(y\,\sharp\,z)=(x\stackrel{{\scriptstyle\leftharpoonup}}{{\cdot}}y)\,\sharp\,z$$

(3)

$$x\stackrel{{\scriptstyle\rightharpoonup}}{{\cdot}}(y\,\sharp\,z)=x\stackrel{{\scriptstyle\rightharpoonup}}{{\cdot}}y\stackrel{{\scriptstyle\rightharpoonup}}{{\cdot}}z$$

(4)

$$(x\,\sharp\,y)\stackrel{{\scriptstyle\leftharpoonup}}{{\cdot}}z=x\stackrel{{\scriptstyle\leftharpoonup}}{{\cdot}}y\stackrel{{\scriptstyle\leftharpoonup}}{{\cdot}}z$$

(5)

for all $x,y,z\in G$ .

We say that a monoid $S$ is 2-cancellative with respect to a generating set $X$ when it has cancellation on monomials of length 2 in the generators in the sense:

For $x\geq x_{0}$ (as just described) let $\sigma_{x}$ be the unique solution for $\sigma\in(\alpha,\beta)$ to the equation

Define the commutator $[x,y]$ and the associator $(x,y,z)$ by:

The associator subloop is $A(Q):=\langle(x,y,z):x,y,z\in Q\rangle$ .

An element $c$ of a loop $Q$ is a weak inverse property (WIP) element iff for all $x\in Q$ ,

Let $W(Q)$ denote the set of all WIP elements of Q.

Define $d$ on $I\times I$ by