We introduce a new regular RELATION δ on a given group G and show that δ is a congurence RELATION on G, with respect to module the commutator subgroup of G. Then we show that the effect of this RELATION on the FUNDAMENTAL RELATION β is equal to the FUNDAMENTAL RELATION γ, and we conclude that, if ρ is an arbitrary strongly regular RELATION on the hypergroup H, then the effect of δ on ρ, results in a strongly regular RELATION on H such that its quotient is an abelian group.

In this paper, we consider the FUNDAMENTAL RELATION b* as the smallest equivalence RELATION on a playgroup P such that the quotient P/b*, the set of all equivalence classes, is a group. The quotient P/b* is called the FUNDAMENTAL group. We give some interesting results about the FUNDAMENTAL groups.

In this paper, an irreversible quantum Brayton heat engine cycle model composed of two isomagnetic processes and two irreversible adiabatic processes is established. The quantum heat engine works with non-interacting spin-1.2 systems and has multi-irreversibilities. By using detailed numerical examples, this paper gives the FUNDAMENTAL optimal RELATIONship of the quantum Brayton heat engine, in a general case and at high temperature limits, and analyzes the effects of internal friction and bypass heat leakage on the FUNDAMENTAL optimal RELATIONship. Comparisons of the power output versus efficiency characteristics among this spin quantum heat Brayton engine, a quantum spin Carnot heat engine and a Brayton heat engine working with a classical working medium, are made. Three special cases, such as the endoreversible case, the frictionless case and the case without bypass heat leakage, are discussed.

In this paper, by considering the notion of -hyperalgebras for an arbitrary signature  , we study the notions of regular and strongly regular RELATIONs on a -hyperalgebra, A. We show that each regular RELATION which contains a strongly regular RELATION is a strongly regular RELATION. Then we concentrate on the connection between the FUNDAMENTAL RELATION of A and the set of complete parts of A.