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Sharp and meager elements in orthocomplete homogeneous effect algebras. (English) Zbl 1193.03084
According to [G. Jenča, Bull. Aust. Math. Soc. 64, No. 1, 81–98 (2001; Zbl 0985.03063)], a homogeneous effect algebra is one in which $$u \leq v_1 \oplus v_2 \leq u'$$ implies that $$u = u_1 \oplus u_2$$ for some $$u_1 \leq v_1$$ and $$u_2 \leq v_2$$. An element $$x$$ of an effect algebra $$E$$ is said to be sharp if $$x \wedge x'= 0$$, and meager if $$0$$ is the single sharp element below $$x$$. If $$E$$ is lattice-ordered and complete, then the set $$S(E)$$ of all its sharp elements is known to be a complete sublattice. The author shows that if $$E$$ is orthocomplete and homogeneous, then the subset $$M(E)$$ of its meager elements forms a commutative BCK-algebra with the relative cancellation property.
The main result of the paper states that a complete lattice effect algebra $$E$$ is, up to isomorphism, characterised by the triple $$(S(E), M(E), h)$$, where $$h$$ is the mapping that associates the lower end $$\{x \in M(E): x \leq a\}$$ of $$M(E)$$ to every $$a \in S(E)$$. The proof depends on the fact that $$E$$ is actually orthocomplete and homogeneous. An explicit construction of a copy of $$E$$, resemblig the triple construction of Stone algebras, is presented. The inverse problem – which triples consisting of a lattice effect algebra $$S$$, BCK-algebra $$M$$ and an appropriate function $$h$$ arise from a complete lattice effect algebra – is not addressed in the paper.

##### MSC:
 03G12 Quantum logic 06C15 Complemented lattices, orthocomplemented lattices and posets 06F35 BCK-algebras, BCI-algebras 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
Zbl 0985.03063
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##### References:
 [1] Bennett, M.K., Foulis, D.J.: Interval and scale effect algebras. Adv. Appl. Math. 19, 200–215 (1997) · Zbl 0883.03048 [2] Cattaneo, G.: A unified framework for the algebra of unsharp quantum mechanics. Int. J. Theor. Phys. 36, 3085–3117 (1997) · Zbl 0898.03022 [3] Chang, C.C.: Algebraic analysis of many-valued logics. Trans. Am. Math. Soc. 88, 467–490 (1959) · Zbl 0084.00704 [4] Chen, C.C., Grätzer, G.: Stone lattices I. Construction theorems. Can. J. Math. 21, 884–894 (1969) · Zbl 0184.03303 [5] Chen, C.C., Grätzer, G.: Stone lattices II. Structure theorems. Can. J. Math. 21, 895–903 (1969) · Zbl 0184.03304 [6] Chevalier, G., Pulmannová, S.: Some ideal lattices in partial abelian monoids. Order 17, 75–92 (2000) · Zbl 0960.03053 [7] Chovanec, F., Kôpka, F.: Boolean D-posets. Tatra Mt. Math. Publ. 10, 183–197 (1997) · Zbl 0915.03052 [8] Dvurečenskij, A.: Effect algebras which can be covered by MV-algebras. Int. J. Theor. Phys. 41, 221–229 (2002) · Zbl 1022.06005 [9] Dvurečenskij, A., Graziano, M.G.: Remarks on representations of minimal clans. Tatra Mt. Math. Publ. 15, 31–53 (1998) · Zbl 0939.06017 [10] Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer and Ister Science, Dordrecht (2000) [11] Foulis, D.J., Bennett, M.K.: Effect algebras and unsharp quantum logics. Found. Phys. 24, 1325–1346 (1994) · Zbl 1213.06004 [12] Foulis, D.J., Greechie, R., Rütimann, G.: Filters and supports in orthoalgebras. Int. J. Theor. Phys. 35, 789–802 (1995) · Zbl 0764.03026 [13] Giuntini, R., Greuling, H.: Toward a formal language for unsharp properties. Found. Phys. 19, 931–945 (1989) [14] Greechie, R., Foulis, D., Pulmannová, S.: The center of an effect algebra. Order 12, 91–106 (1995) · Zbl 0846.03031 [15] Gudder, S.: Sharply dominating effect algebras. Tatra Mt. Math. Publ. 15, 23–30 (1998) · Zbl 0939.03073 [16] Jenča, G.: Subcentral ideals in generalized effect algebras. Int. J. Theor. Phys. 39, 745–755 (2000) · Zbl 0957.03060 [17] Jenča, G.: Blocks of homogeneous effect algebras. Bull. Aust. Math. Soc. 64, 81–98 (2001) · Zbl 0985.03063 [18] Jenča, G.: Finite homogeneous and lattice ordered effect algebras. Discrete Math. 272, 197–214 (2003) · Zbl 1031.03078 [19] Jenča, G., Pulmannová, S.: Quotients of partial abelian monoids and the Riesz decomposition property. Algebra Univers. 47, 443–477 (2002) · Zbl 1063.06011 [20] Jenča, G., Pulmannová, S.: Orthocomplete effect algebras. Proc. Am. Math. Soc. 131, 2663–2671 (2003) · Zbl 1019.03046 [21] Jenča, G., Riečanová, Z.: On sharp elements in lattice ordered effect algebras. BUSEFAL 80, 24–29 (1999) [22] Kôpka, F., Chovanec, F.: D-posets. Math. Slovaca 44, 21–34 (1994) [23] Mundici, D.: Interpretation of AF C *-algebras in Lukasziewicz sentential calculus. J. Funct. Anal. 65, 15–53 (1986) · Zbl 0597.46059 [24] Mundici, D.: MV-algebras are categorically equivalent to bounded commutative BCK-algebras. Math. Jpn. 6, 889–894 (1986) · Zbl 0633.03066 [25] Pták, P., Pulmannová, S.: Orthomodular Structures as Quantum Logics. Kluwer, Dordrecht (1991) · Zbl 0743.03039 [26] Riečanová, Z.: A generalization of blocks for d-lattices and lattice effect algebras. Int. J. Theor. Phys. 39, 231–237 (2000) · Zbl 0968.81003 [27] Riečanová, Z.: Continuous lattice effect algebras admitting order continuous states. Fuzzy Sets Syst. 136, 41–54 (2003) · Zbl 1022.03047 [28] Yutani, H.: The class of commutative BCK-algebras is equationally definable. Math. Semin. Notes 5, 207–210 (1977) · Zbl 0373.02045
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