### A note on equalities of radicals in a semigroup

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On any involuted semigroup $(S,\xb7{,}^{\text{'}})$, define the ternary operation $\left[abc\right]:=a\xb7{b}^{\text{'}}\xb7c$ 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...

Semiheaps are ternary generalisations of involuted semigroups. The first kind of semiheaps studied were heaps, which correspond closely to groups. We apply the radical theory of varieties of idempotent algebras to varieties of idempotent semiheaps. The class of heaps is shown to be a radical class, as are two larger classes having no involuted semigroup counterparts. Radical decompositions of various classes of idempotent semiheaps are given. The results are applied to involuted I-semigroups, leading...