Nell’ambito del programma "Visiting Professors", sponsorizzato dalla Regione Sardegna, i proff. Anatolij Dvurecenskij (Slovak Academy of Sciences) e Tomasz Kowalski (Australian National University / Università di Cagliari) terranno due seminari:

 

Facoltà di Scienze della Formazione

Aula 1 corpo centrale

 

We show what a state can be in different algebraic structures like quantum structures, Hilbert space quantum logic, OML, OPM, effect algebras, MV-algebras, BL-algebras as well in noncommutative structures like GMV-algebras. We show a relationship to de Finetti coherence principle and we mention state-morphism MV-algebras.

 

On an (old) finite basis problem in loop theory Abstract Any Latin square L can be thought of as a multiplication table. Such multiplication will typically be nonassociative, but it has unique solution property and by permuting some rows and columns we can always make it have a unit. So massaged L becomes a loop. Formally, a loop is a groupoid with unique solution property, but with a nonassociative multiplication. A loop L is a group if and only if L is associative.

One effect of nonassociativity of L is that powers of an element a are not well-defined: a^3 is ambiguous between (a*a)*a and a*(a*a), or equivalently, not every one-generated subloop of L is a group.

Consider the class of all loops such that every one-generated subloop is a group. Such loops are called power-associative, and they can be axiomatised by an

(infinite) set of equations, so they form a subvariety of loops.

Evans and Neumann [1] showed that this infinite set cannot be replaced by any finite set of equations, so the variety of power-associative loops is not finitely based. A natural midpoint between power-associative loops and groups is the variety of diassociative loops: such that every two-generated subloop is a group.

Evans and Neumann considered these as well and asked whether they are finitely based. A negative solution was proposed in [2] but found wanting. In [3] the problem was still reported as open "although everyone believes that the solution is in the negative". A recent [4] proposed another negative solution relying on some deep results in number theory, but it remains unverified.

We will show that diassociativity is not finitely based over power-associativity. Our result is stronger than the one announced in [4] and the proof technique is different: we use elementary constructions and one deep result from model theory.

 

 

[1] T. Evans, B.H. Neumann, On varieties of groupoids and loops, J. London Math. Soc. 28 (1953) 342-350.

[2] D.M. Clark, Diassociative groupoids are not finitely based, J. Australian Math. Soc. 11 (1970) 113-114.

[3] M.K. Kinyon, K. Kunen, J.D. Phillips, A generalisation of Moufang and Steiner loops, Algebra Universalis 48, 1 (2002) 81-101.

[4] W.D. Smith, Loop diassociativity has no finite basis, unpublished manuscript (2005).

 

Francesco Paoli

Associate Professor of Logic

Dept. of Education, University of Cagliari Via Is Mirrionis 1, 09123 Cagliari, Italy Tel. No. 070.6757330 E-mail: paoli@unica.it