Two aspects of the theory of quasi-mv algebras

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1 Two aspects of the theory of quasi-mv algebras Roberto Giuntini, Francesco Paoli, Antonio Ledda, Hector Freytes Department of Education, University of Cagliari Via Is Mirrionis 1, Cagliari (Italy) Corresponding author: F. Paoli, January 22, 2007

2 Contents 0.1 The lattice of subvarieties of quasi-mv algebras Categorical equivalences for p 0 quasi-mv algebras Quasi-MV algebras were introduced in [5] in an attempt to describe an appropriate abstract counterpart of the algebra of all density operators of C 2, endowed with operations corresponding to a few signi cant quantum logical gates. This variety of algebras, apart from its original motivation related to quantum computation, has an intrinsic interest as a generalisation of MV algebras to the semisubtractive but not point regular case. Quasi-MV algebras were later expanded in [3] by an operation of square root of the inverse ( p 0 ) whose quantum computational signi cance is especially noteworthy. In this talk we discuss two aspects of the theory of quasi-mv algebras: the structure of the lattice of subvarieties of quasi-mv algebras and a connection between p 0 quasi-mv algebras and a class of partially ordered groups with additional operators. We refer the reader to the mentioned papers for the necessary background concerning terminology and notation. 0.1 The lattice of subvarieties of quasi-mv algebras The structure of the lattice of subvarieties of MV is well-known, thanks to a comprehensive study due to Komori, as well as to Di Nola and Priestly ([4]; [2]; see also [1]). We intend to investigate as accurately as possible the structure of the lattice L V (QMV) of subvarieties of QMV, against the background of Komori s classi cation of MV algebraic varieties. For a start, the structure of its " at" side is easily described. In fact: Lemma 1 There are just two nontrivial varieties of at QMV algebras: FQMV =V(F 02 ); V(F 10 ), which is axiomatised relative to FQMV by the equation x x 0. The next lemma identi es two splitting pairs in L V (QMV). 1

3 Lemma 2 The pairs hv (F 10 ) ; MVi and hba; FQMVi split the lattice L V (QMV). Corollary 3 V (F 10 ) and BA are the only atoms of L V (QMV). Corollary 4 If V is a variety of quasi-mv algebras, V MV i V is subtractive w.r.t. 0. It turns out that the class of congruence lattices of members of any "pure" subvariety of QMV algebras validates just the equations satis ed by all lattices, and nothing else. More precisely: Theorem 5 If V is a variety of quasi-mv algebras, the following are equivalent: V MV; There is a nontrivial lattice equation which is satis ed in fc(a) : A 2Vg. Indeed, even something stronger holds: V MV i there exists some property P which is satis ed in fc(a) : A 2Vg and implies some nontrivial universal formula in the language of lattices. Therefore, no "genuine" variety of QMV algebras is either congruence distributive, or congruence modular, or congruence permutable, or e-regular. Theorem 55 in [5] implies that the varietal join MV _ FQMV in L V (QMV) is just QMV. What about the binary joins V _ FQMV, where V is a proper subvariety of MV? With the next theorem, we provide a general answer. We have the following result: Theorem 6 Let V be a proper nontrivial subvariety of MV algebras whose equational basis relative to MV is E. The following subvarieties of QMV are mutually coincident: V (fa : A= 2 V SI g) ; V _ FQMV; mod(e); V (fa F 02 : A 2 V SI g). Some special instances of the previous theorem are worth emphasising: for example, BA _ FQMV coincides with mod(x e x 0 0), or with V (B 2 F 02 ), or else with the variety generated by all the A s such that A= = B 2. Likewise, MV n _FQMV coincides with mod((n)x (n 1)x), or with V (f j F 02 : j ng), or else with the variety generated by all the A s such that A= = n. 2

4 0.2 Categorical equivalences for p 0 quasi-mv algebras Daniele Mundici established a well-known equivalence between the categories of MV algebras and Abelian `-groups with strong unit via an invertible functor (the Gamma functor: [1]). A partial analogue of Mundici s Gamma functor turns out to be available in the present framework too, its upshot being a categorical equivalence between pair algebras and a special category of Abelian partially ordered groups with additional operators. We begin by introducing a class of Abelian partially ordered groups endowed with two operators of projection and rotation. De nition 7 An Abelian projection-rotation group (for short, PR-group) is a rst order structure G = hg; +; ; P; R; 0; i, of type hh2; 1; 1; 1; 0i ; h2ii, such that: G1 hg; +; G2 hg; +; ; 0; i is an Abelian po-group; ; P; R; 0i is an Abelian group with operators; G3 P (G) and P R(G) are lattice ordered subgroups of G, and G is the direct sum of (isomorphic copies of) such; P1-P4 for all a 2 G: P a = P a; P P a = P a; P (P a ^ P b) = P a ^ P b and P (P a _ P b) = P a _ P b; if a b, then P a P b: R1-R2 for all a 2 G: R a = Ra; RRa = a. PR1-PR3 for all a; b 2 G, P RP a = 0; P R(P a ^ P b) = P RP a ^ P RP b and P R(P a _ P b) = P RP a _ P RP b; for all a; b 2 G, if a b, then P Ra P Rb. The de nition of strong order unit which is usually given for partially ordered groups can be adapted in such a way as to guarantee an appropriate interaction with the operators P and R. De nition 8 Let G = hg; +; ; P; R; 0; i be an Abelian PR-group. A positive element u 2 G is said to be a strong order unit of G just in case: (U1) for all a 2 G, there exists a nonnegative integer n s.t. a nu; (U2) P u u; (U3) for all a 2 G, if u a u, then u Ra u. An important example of Abelian PR-group is the po-group of the complex numbers, where P and R are, respectively, the projection operator onto the X axis and the 2 clockwise rotation operator; (1; 1) is a strong order unit for that group. Further signi cant examples (e.g. over the po-group of bounded functions of one real variable, or over the underlying po-groups of nite-dimensional inner product vector lattices) can be provided. 3

5 De nition 9 Let G = G; + G ; G ; P G ; R G ; 0 G ; G be an Abelian PR-group with strong order unit u. The interval algebra i (G; u) in G is the algebra [ u; u] ; i(g;u) ; p 0 i(g;u) ; 0 i(g;u) ; 1 i(g;u) ; k i(g;u) of type h2; 1; 0; 0; 0i, such that, omitting unnecessary superscripts, a b = P (a + b + u) ^ P u; p 0 a = Ra; 0 i(g;u) = P u; 1 = P u; k = 0 G. Theorem 10 Let G = G; + G ; G ; P G ; R G ; 0 G ; G be an Abelian PR-group with p strong order unit u. i) The interval algebra i(g; u) in G is a cartesian 0 QMV algebra; ii) i(g;u) = G d[ u; u]. To invert the preceding functor, the rst ingredient we need is the following de nition. De nition 11 Let G = hg; +; ; 0; i be an Abelian `-group. Its Gaussian square is the structure D G(G) = G 2 ; + G(G) ; G(G) ; P G(G) ; R G(G) ; 0 G(G) ; G(G)E, where G 2 ; + G(G) ; G(G) ; 0 G(G) ; G(G) is the direct product G G and, for a; b 2 G, P G(G) ha; bi = a; 0 G and R G(G) ha; bi = b; G a. We immediately have that: Lemma 12 (i) The Gaussian square G(G) of an Abelian `-group G is an Abelian PR-group; (ii) if the former has a strong unit u G, the latter has a strong unit u G ; u G ; (iii) G is embeddable into the appropriate reduct of G(G). Recall from [1] that, whenever M is an MV algebra, the structure (M) of all equivalence classes of pairs of good sequences of M, endowed with appropriate operations, is an Abelian `-group with strong order unit [(1) ; (0)]. This applies in particular to the MV algebra R Q of regular elements of a cartesian p 0 QMV algebra. As a consequence of the previous lemma, we have then: Corollary 13 Let Q = DQ; Q ; p E 0 Q ; 0 Q ; 1 Q ; k Q be a cartesian p 0 QMV algebra. The Gaussian square i (Q) = G ((R Q )) is an Abelian PR-group with strong unit u (R Q) ; u (R Q). Theorem 14 Let Q = DQ; Q ; p E 0 Q ; 0 Q ; 1 Q ; k Q be a cartesian p 0 QMV algebra. Then Q is embeddable into i ( i (Q)). 4

6 An interesting consequence of (the proof of) the preceding result is a categorical equivalence between pair algebras - i.e. p 0 QMV algebras Q s.t. Q 'P (R Q ) - and Abelian PR-groups. Since the former category can be proved equivalent to the category of MV algebras, it follows that the following categories are mutually equivalent: MV algebras; Pair algebras; Abelian `-groups with strong unit; Abelian PR-groups. 5

7 Bibliography [1] Cignoli R., D Ottaviano I.M.L., Mundici D., Algebraic Foundations of Many- Valued Reasoning, Kluwer, Dordrecht, [2] Di Nola A., Priestly A.H.A., "Equational characterization of all varieties of MV algebras", Journal of Algebra, 221, 2, 1999, pp [3] Giuntini R., Ledda A., Paoli F., "Expanding quasi-mv algebras by a quantum operator", Studia Logica, forthcoming; available from [4] Komori Y., "Super ukasiewicz propositional logics", Nagoya Mathematical Journal, 84, 1981, pp [5] Ledda A., Konig M., Paoli F., Giuntini R., "MV algebras and quantum computation", Studia Logica, 82, 2, 2006, pp