We say that a variety $V$ is reversible if for each $n>0$ and $n$-ary fundamental operation $f$, there is some $m\geq n$ and $r$ along with terms $T_{2},\dots,T_{r},S_{1},\dots,S_{m}$ such that $G,H$ are inverses where

$G(x_{1},\dots,x_{m})=(f(x_{1},\dots,x_{n}),T_{2}(x_{1},\dots,x_{m}),\dots,T_{r}(x_{1},\dots,x_{m})),$ and

$H(y_{1},\dots,y_{r})=(S_{1}(y_{1},\dots,y_{r}),\dots,S_{m}(y_{1},\dots,y_{r})).$

The following varieties are reversible:

Groups.

Racks,quandles, and biracks.

Jonsson-Tarski algebras.

The variety consisting of all algebras $(X,f,g)$ where $f,g:X\rightarrow X$ are inverse functions.

The variety of all heaps.

The variety of quasigroups.

The variety $K_{X}$ where $K_{X}$ is generated by the algebra $(X,\mathcal{F})$ where $\mathcal{F}$ consists of all functions $f:X^{n}\rightarrow X$ where $|f^{-1}[\{a\}]|=|f^{-1}[\{b\}]|$ for each $a,b\in X$.

The variety consisting of sets without any fundamental operations.

In addition to these varieties, one can artificially create (hopefully non-trivial) reversible varieties since the fact that $G,H$ are inverses is clearly axiomatized by identities. Also, every subvariety of a reversible variety is reversible. It is therefore quite easy to contrive new reversible varieties from old ones.

If $X$ is a finite algebra in a reversible variety and $f$ is an $n$-ary fundamental operation and $a\in X$, then $$|\{(x_{1},\dots,x_{n})\in X^{n}\mid f(x_{1},\dots,x_{n})=a\}|=|X|^{n-1}.$$

Observe that the reversible varieties are the varieties consisting of algebras that are “almost groups” in the sense that one often associates these algebras with canonical groups and because these algebras feel like groups.

What are some more examples of reversible varieties that occur naturally and in practice and are not just constructed for the sole purpose of being reversible? Is there a good reference for reversible varieties?