A. Ledda T. Kowalski F. Paoli On Certain Quasivarieties of Quasi-MV Algebras Abstract. Quasi-MV algebras are generalisations of MV algebras arising in quantum computational logic. Although a reasonably complete description of the lattice of sub- varieties of quasi-MV algebras has already been provided, the problem of extending this description to the setting of quasivarieties has so far remained open. Given its appar- ent logical repercussions, we tackle the issue in the present paper. We especially focus on quasivarieties whose generators either are subalgebras of the standard square quasi- MV algebra S, or can be obtained therefrom through the addition of some fixpoints for the inverse. Keywords : quasi-MV algebras, quasivarieties, MV-algebras. Introduction Quasi-MV algebras (for short, qMV algebras) were introduced in [8] in con- nection with quantum computational logic — namely, in an attempt to pro- vide a convenient abstraction of the algebra over the set of all density op- erators of the Hilbert space C 2 , endowed with a suitable stock of quantum logical connectives. Independently of their original quantum computational motivation, qMV algebras present an additional, purely algebraic, motive of interest as generalisations of MV algebras to the quasi-subtractive (in the sense of [7]) but not point regular case. In subsequent papers ([9], [1], [5], [2]) the algebraic properties of qMV algebras and their associated logics were investigated in greater detail. The above referenced papers contain the ba- sics of the structure theory for this variety, including appropriate standard completeness theorems w.r.t. the algebras over the complex numbers which constituted the motivational starting point of the whole investigation. In [1], in particular, a reasonably complete description of the lattice L V (qMV) of varieties of qMV algebras was provided. However, the prob- lem of extending this description to the setting of quasivarieties remained open. Given its apparent logical repercussions, we will tackle the issue in the present paper. The results we achieve are not just as satisfactory, because Special Issue: Algebras Related to Non-classical Logic Edited by Manuel Abad and Alejandro Petrovich Studia Logica (2011) 98: 145–170 c  Springer 2011