On any involuted semigroup $(S,·{,}^{\text{'}})$, define the ternary operation $\left[abc\right]:=a·b^{\text{'}}·c$ for all $a,b,c\in S$. The resulting ternary algebra $(S,[\left]\right)$ satisfies the para-associativity law $\left[\right[abc\left]de\right]=\left[a\right[dcb\left]e\right]=\left[ab\right[cde\left]\right]$, which defines the variety of semiheaps. Important subvarieties include generalised heaps, which arise from inverse semigroups, and heaps, which arise from groups. We consider the intermediate variety of near heaps, defined by the additional laws $\left[aaa\right]=a$ and $\left[aab\right]=\left[baa\right]$. Every Clifford semigroup is a near heap when viewed as a semiheap, and we show that the Clifford semigroup...

The class of semigroups satisfying semimedial laws is studied. These semigroups are called semimedial semigroups. A connection between semimedial semigroups, trimedial semigroups and exponential semigroups is presented. It is proved that the class of strongly semimedial semigroups coincides with the class of trimedial semigroups and the class of dimedial semigroups is identical with the class of exponential semigroups.