D-homomorphisms and atomic ±complete boolean D-posets

We study observables as r-D-homomorphisms de®ned on Boolean D-posets of subsets into a Boolean Dposet. We show that given an atomic r-complete Boolean D-poset P with the countable set of atoms there exist a r-complete Boolean D-poset of subsets S and a r-D-homomorphism h from S onto P, more precisely we can choose S ˆ B…R†, which gives an analogy of the Loomis± Sikorski representation theorem for Boolean r-algebras. We show also that any atomic r-complete Boolean D-poset with the countable set of atoms is the range of a r-homomorphism de®ned on a r-complete Boolean D-poset of fuzzy sets which gives another type of the Loomis±Sikorski theorem. Key words D-poset, Boolean D-poset, D-homomorphism, observable, tribe, homomorphism 1 Introduction A new algebraic structure, called a difference poset (shortly a D-poset), has been recently introduced as an axiomatic model for foundations of quantum mechanics. The idea of difference posets is very simple. If we have two comparable events a and b (a b), then our knowledge of a and b entails the complete knowledge of the rest of a in b, i.e. b a. First this idea was applied to fuzzy set problems in quantum mechanics [14] and then it was presented in a general algebraic form ([16]). Later on, some other algebraic structures have been introduced (RI-sets [13], D-sets [18], which generalize difference posets [11] and D-algebras [10]), in which a difference is a primary operation. D-posets generalize orthomodular posets (= quantum logics) [19], orthoalgebras [20] and even the set of all effects [6], which are important for modelling unsharp measurement in quantum mechanics. Other proposed models of the class of all effects are weak orthoalgebras [9] and effect algebras [8]. Although these frameworks are algebraically equivalent, they originated due to completely different starting points and they have their original systems of axioms. The similar situation can be seen in the theory of in®nite-valued (èukasiewicz) logics, where Wajsberg algebras [7] and MV-algebras [1] are the same things. F. KoÃpka [15] studied the compatibility in D-posets and de®ned a Boolean D-poset. A Boolean D-poset can be characterized as a difference lattice of pairwise compatible elements. (We note that an orthomodular lattice of pairwise compatible elements is a Boolean algebra). By [3], every MV-algebra is a Boolean D-poset and, conversely, a Boolean D-poset can be organized as an MV-algebra. We recall that our partial binary operation de®nes a total binary operation ‡ via a‡ b ˆ 1 ……1 a† ……1 a† ^ b†† : In this case, ‡ is a basic MV-algebra operation. The results of a special direction in MV-algebras and D-posets research, especially of probability theory on these structures, can be found in [21], [12]. Our approach is motivated by the problems of quantum structures, and therefore, we use also a language of quantum structures. The basic notion is an observable. That corresponds to the requirement that if A B then measured quantities x…A† and x…B† must be ordered, i.e., x…A† x…B†, and measurement of B n A is completely determined by values of x…A† and x…B†, i.e. x…B n A† ˆ x…B† x…A†. This gives a so called D-homomorphism of D-posets that preserves and constant elements 0 and 1. In the case of a classical quantum structure, a Boolean r-algebra of subsets of a set X, this notion of an observable is equivalent to a random variable n : X! R via x ˆ nÿ1. In the present paper, we shall study atomic r-complete Boolean D-posets and r-D-homomorphisms. We show that given an atomic r-complete Boolean D-poset P with the countable set of atoms there exist an atomic r-complete Boolean D-poset of subsets S of a set X 6ˆ ; and a r-Dhomomorphism h fromS onto P. In addition, we can show that this D-homomorphism is in fact an observable de®ned on B…R†. This result is a kind of the Loomis ± Sikorski theorem for Boolean r-algebras [22]. Another kind of this representation will be presented showing that any atomic r-complete Boolean D-poset with the countable set of atoms is a r-homomorphic image of some r-complete Boolean D-poset of fuzzy sets (= a tribe of fuzzy sets). Original paper Soft Computing 4 (2000) 9±18 Ó Springer-Verlag 2000