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Compressible effect algebras. (English) Zbl 1075.81011
The author sets out to vastly generalize results from D. J. Foulis’ papers on “Compressions on partially ordered abelian groups” [Proc. Am. Math. Soc. 132, 3581–3587 (2004; Zbl 1063.47003)], and on “Compressible groups” [Math. Slovaca 53, 433–455 (2003; Zbl 1068.06018)] to arbitrary effect algebras $$(E,0,1,\oplus)$$. By $$a'$$ one denotes the supplement of $$a\in E$$, i. e. $$a\oplus a'=0$$. If $$a\wedge a' = 0$$ then $$a$$ is called sharp, $$E_s$$ denotes the set of sharp elements of $$E$$. A retraction $$J$$ is an additive map $$E \rightarrow E$$ such that $$a\leq J(1)$$ implies $$J(a)=a$$. $$J(1)$$ is called the focus of $$J$$. If $$J(a)=0$$ implies $$a\leq J(1)'$$ a retraction $$J$$ is called a compression. A retraction $$J$$ is called direct if $$J(a)\leq a$$ for all $$a\in E$$. Direct retractions $$J$$ induce an isomorphism between $$E$$ and the cartesian product algebra $$\text{Ker}(J)\oplus J[E]$$. Thus, they are equivalent to projections of a cartesian product onto a factor (Thm. 3.1, 3.2). If in an effect algebra $$E$$ every compression is uniquely determined by its focus $$p$$ , and has a supplement, then $$E$$ is called compressible. A finite cartesian product of effect algebras is compressible iff every component is compressible, whereas an analogous statement for horizontal sums does not hold.
If $$p\in E$$ is the focus of some compression, then it is called a projection; $$P$$ denotes the set of all projections in $$E$$. In compressible effect algebras compressions may be uniquely denoted by $$J_p$$ where $$p\in P$$ is chosen such that $$J_p(1)=p$$. Using the binary operation $$\circ$$ on $$P\times E$$ defined by $$p\circ a = J_p(a)$$, an element $$a\in E$$ is called compatible with $$p$$ if $$a= (p\circ a) \oplus (p'\circ a)$$, and $$C(p)= \{a\in E: a= (p\circ a) \oplus (p'\circ a)\}$$ is called the commutant of the projection $$p$$. The operation $$\circ$$ is examined in some detail. Among others the author proves for $$p,q\in P$$: $$p\circ q = q\circ p$$ iff $$p\in C(q)$$ iff $$p\circ q \in P$$. $$P$$ is a sub-effect algebra of $$E$$ and an orthomodular poset. For the commutant of $$p$$ one has $$C(p)=E$$ iff $$J_p$$ is direct. The projection commutant $$C_p(a):=\{p\in P : a \in C(p)\}$$ is a sub-orthomodular poset of $$P$$.
A sequential effect algebra (SEA) is an effect algebra equipped with an operation $$\circ$$ on $$E\times E$$ such that for all $$a,b,c \in E$$ one has that $$b\rightarrow a\circ b$$ is additive, $$1\circ a = a$$, $$a\circ b = 0$$ implies $$b\circ a =0$$, $$a\circ(b\circ c) = (a \circ b)\circ c$$ if $$a$$ and $$b$$ commute, and some further mild demand of commutativity. A SEA may be incompressible. In a compressible SEA $$E$$ the restriction of $$\circ$$ to $$P\times E$$ equals $$(p,a)\rightarrow J_p(a)$$. Insofar, a compressible effect algebra may be imagined as part of a SEA. (5.1) gives $$E_s=P$$, $$a\in C(p)$$ iff $$p\circ a = a \circ p$$, (5.2) characterizes the compressibility of a SEA $$E$$ by the property that every retraction $$J$$ pf $$E$$ with focus $$p$$ is given by $$J(a)=p\circ a, p\in E_s$$.
If $$E$$ is compresssible or a SEA with the projection-cover property, then $$P(E)$$ is an orthomodular lattice. Further items are Lüders maps and conditional probabilities.

MSC:
 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) 03G12 Quantum logic 06C15 Complemented lattices, orthocomplemented lattices and posets 05E25 Group actions on posets, etc. (MSC2000) 62L99 Sequential statistical methods 06D35 MV-algebras
Citations:
Zbl 1063.47003; Zbl 1068.06018
Full Text:
References:
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