# Examples of associative inducers and other inducers

I am curious about how well the following technique can produce algebraic structures and semigroups in particular.

Let $$(X,\circ)$$ be a semigroup. Let $$Y$$ be a set and let $$L:X\rightarrow P(Y)$$ be a function. Let $$T:\{(x,y,z)|x,y\in X,z\in L(x\circ y)\}\rightarrow Y\times\{0,1\}$$ be a function where if $$T(x,y,z)=(s,0)$$, then $$s\in L(x)$$ and if $$T(x,y,z)=(s,1)$$, then $$s\in L(y)$$.

Let $$A$$ be a set and define an operation $$\circ^{A}_{T}$$ on $$\bigcup_{x\in X}\{x\}\times A^{L(x)}$$ by letting $$(x,f)\circ^{A}_{T}(y,g)=(x\circ y,h)$$ precisely when $$h(z)= \begin{cases} f(s) & \text{if there is some s with} & T(x,y,z)=(s,0) \\ g(s) & \text{if there is some s with} & T(x,y,z)=(s,1). \end{cases}$$

We shall call the operation $$T$$ an inducer. If $$\circ_{T}^{A}$$ is associative for all sets $$A$$, then we shall say that the operation $$T$$ is an associative inducer.

For example, if $$\lambda$$ is a limit ordinal and $$L:\lambda\rightarrow P(\lambda)$$ is the function where $$L(\alpha)=\alpha$$, then $$(\lambda,+)$$ is an associative operation and if we define $$T:\{(\alpha,\beta,\gamma):\alpha,\beta<\lambda,\gamma<\alpha+\beta\}\rightarrow\lambda\times\{0,1\}$$ by letting $$T(\alpha,\beta,\gamma)=(\gamma,0)$$ whenever $$\gamma<\alpha$$ and $$T(\alpha,\beta,\alpha+\gamma)=(\gamma,1)$$ whenever $$\gamma<\beta$$, then $$T$$ is an associative inducer and $$\circ^{A}_{T}$$ is simply the transfinite concatenation operation.

The multigenic and endomorphic Laver tables give highly non-trivial and combinatorially complex examples of associative inducers.

What are some other examples of associative inducers? Is there a good reference for this notion?

Observe by associating functions $$L:X\rightarrow P(Y)$$ with relations $$R\subseteq X\times Y$$, that the notion of an associative inducer is first order axiomatizable by universal formulas.

The notion of an inducer makes sense for other varieties besides the variety of semigroups. For example, the multigenic and endomorphic Laver tables produce self-distributive inducers. I am interested in the inducers that produce algebras in other varieties besides the variety of semigroups as well.