Pseudo-hoops. (English) Zbl 1078.06007

B. Bosbach’s pioneering papers on various classes of semigroups were largely unnoticed: among others he introduced “complementary semigroups” (today known as pseudo-hoops): a deep investigation of these structures is found in his papers “Komplementäre Halbgruppen. Axiomatik und Arithmetik” [Fundam. Math. 64, 257–287 (1969; Zbl 0183.30603)] and “Komplementäre Halbgruppen. Kongruenzen and Quotienten” [Fundam. Math. 69, 1–14 (1970; Zbl 0263.20037)]. Bosbach even found an equivalent reformulation of MV-algebras, in his paper “Concerning bricks” [Acta Math. Acad. Sci. Hung. 38, 89–104 (1981; Zbl 0533.06009)]. In section 1.2, building on Bosbach’s work, the authors of the present paper pursue the investigation of pseudo-hoops. These structures include, among others, “noncommutative” generalizations of Hajek’s BL-algebras, whence of MV-algebras. Various subclassess are investigated, including cancellative, product, and basic pseudo-hoops. The latter are shown to be equivalent to pseudo-BL-algebras. It is shown that bounded Wajsberg pseudo-hoops are equivalent to pseudo-Wajsberg algebras. In a final section, examples are given of pseudo-hoops and their normal filters.


06F05 Ordered semigroups and monoids
03G25 Other algebras related to logic
06D35 MV-algebras


Pseudo Hoops