The lattice of subvarieties of √ ′ quasi-MV algebras Tomasz Kowalski, Francesco Paoli, Roberto Giuntini, Antonio Ledda Department of Education, University of Cagliari October 14, 2009 Abstract In the present paper we continue the investigation of the lattice of subvarieties of the variety of √ ′ quasi-MV algebras, already started in [6]. Beside some general results on the structure of such a lattice, the main contribution of this work is the solution of a long-standing open problem concerning these algebras: namely, we show that the variety generated by the standard disk algebra Dr is not finitely based, and we provide an infinite equational basis for the same variety. 1 Introduction √ ′ quasi-MV algebras (for short, √ ′ qMV algebras) were introduced in [3] as an axiomatisation of the equational theory of the algebra of density operators of the Hilbert space C 2 , endowed with operations corresponding to the quan- tum computational connectives of quantum  Lukasiewicz disjunction and square root of negation : see e.g. [5] for a general motivational introduction and for a thorough explanation of the connection between these structures and quantum computation. Over and above their relevance to physics and quantum logic, √ ′ qMV algebras are of interest to the fuzzy logician as expansions of generalisa- tions of Chang’s MV algebras [2]. The structure theory of this variety has been intensively investigated over the past few years: see [9], [1], [4], [6]. In partic- ular, its ”well-behaved” proper subquasivariety of Cartesian √ ′ qMV algebras turned out to be special in that its members can be represented as subalgebras of intervals in Abelian ℓ-groups with additional operators ([4]). An investigation of the lattice of √ ′ qMV varieties was started in [6]. Here we intend to round off its description by providing additional results on the topic. In detail, the paper is organised as follows. In Section 2 we recap the results obtained in [6] on the lattice of √ ′ qMV varieties. In Section 3 we somehow stray away from our main thread by proving a few lemmas concerning MV algebras, justified by their subsequent use in the remainder of the paper. Section 4 contains the main results of the article, including a proof of the fact that the variety generated by the standard disk algebra D r is not finitely based, 1