Rectangular band ($x\ast x=x,(x\ast y)\ast (z\ast w)=x\ast w$)

A rectangular band is a semigroup satisfying the displayed identities; these also imply associativity. Every nonempty such algebra is a product $B\times C$ with $(b,c)\ast (b^{\prime },c^{\prime })=(b,c^{\prime })$. Fixing $e$, the factor sets are $B=\{x\ast e\}$ and $C=\{e\ast x\}$, and $x\mapsto (x\ast e,e\ast x)$ has inverse $(b,c)\mapsto b\ast c$. Homomorphisms between nonempty products are exactly pairs of maps on the two factors.

The category of rectangular bands is Cartesian closed. For nonempty products $A=B\times C$, $D=E\times F$, the exponential is $E^B\times F^C$. Also $D^0=1$ and $0^A=0$ for nonempty $A$. It has no subobject classifier: the classifier would have two elements, hence would be left-zero or right-zero, while its characteristic maps on a two-by-two product could not single out a one-point subalgebra.