Compressible effect algebras.

*(English)*Zbl 1075.81011The 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.

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.

Reviewer: Horst Szambien (Garbsen)

##### 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 |

##### Keywords:

effect algebra; compressible; sequential product; projection; retraction; orthomodular poset; orthomodular lattice; Lüders map
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