Minimal quasi-ideals in ternary semigroup (Q1366918)

Assume the ternary semigroup \((T,[\cdots])\) without \(0\) has at least one ternary idempotent element. A non-empty subset \(Q\) of \(T\) is called a quasi-ideal iff \([TTQ]\cap\{[TQT]\cup[TTQTT]\}\cap[QTT]\subset Q\). All minimal quasi-ideals of a given semigroup are isomorphic. Moreover, every minimal quasi-ideal \(Q\) of \(T\) is a ternary subgroup of \(T\) and can be written in the form \(Q=[eTeTe]\), where \(e\) is the identity of \(Q\). If \(T\) has at least one minimal quasi-ideal, then the union of all minimal quasi-ideals of \(T\) is the kernel of \(T\) in the sense of \textit{F. M. Sioson} [Math. Jap. 10, 63-84 (1965; Zbl 0247.20085)].