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Perfect effect algebras are categorically equivalent with Abelian interpolation po-groups

Anatolij Dvurečenskij

2007
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Journal of the Australian Mathematical Society
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We introduce perfect effect algebras and we show that every perfect algebra is an interval in the lexicographical product of the group of all integers with an Abelian directed interpolation po-group. To show this we introduce prime ideals of effect algebras with the Riesz decomposition property (RDP). We show that the category of perfect effect algebras is categorically equivalent to the category of Abelian directed interpolation po-groups. Moreover, we prove that any perfect effect algebra is
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... subdirect product of antilattice effect algebras with the RDP. 2000 Mathematics subject classification: primary 06F20; secondary 03G12, 03B50. Keywords and phrases: effect algebra, the Riesz decomposition property, radical, perfect effect algebra, interpolation po-group, unital po-group, categorical equivalence. 183 use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700016025 Downloaded from https://www.cambridge.org/core. IP address: 207.241.231.80, on 23 Jul 2018 at 02:36:31, subject to the Cambridge Core terms of Anatolij Dvurecenskij [2] effect algebra with the Riesz decomposition property. Such algebras are always intervals in unital ^-groups. Motivated by MV-algebras, we introduce perfect effect algebras. In such algebras, every element is either in its radical or in its logical complement. Such algebras are always intervals in the lexicographical product of the group of all integers with an Abelian directed interpolation po-group. Moreover, we show that we have a categorical equivalence of the category of perfect effect algebras and the category of Abelian interpolation directed po-groups. This generalizes an analogous result of Di Nola and Lettieri [2] for perfect effect algebras. The paper is organized as follows. Effect algebras are presented in Section 2 with the main emphasis on the Riesz decomposition property. Ideals of effect algebras (maximal ideals, prime ideals and values) are studied in Section 3. In Section 4, we introduce infinitesimals of effect algebras and the radical, the most important part of every effect algebra. We show that in contrast to MV-algebras, not every radical consists of all its infinitesimals. Therefore, we introduce effect algebras with the Rad-property. Perfect effect algebras are introduced in Section 5, where we show the categorical equivalence of the category of perfect effect algebras with the category of Abelian directed interpolation po-groups. In Section 6, we study quotient effect algebras, and in Section 7 we prove that every perfect effect algebra is a subdirect product of antilattice effect algebras with the RDP. Effect algebras with the Riesz decomposition property An effect algebra is a partial algebra E -(E; +, 0, 1) with a partially defined operation + and two constant elements 0 and 1 such that, for all a, b, c e E, (i) a+b is defined in E if and only if b+a is defined, and in this case a+b = b+a; (ii) a + b, (a + b) + c are defined if and only if b + c and a + (b + c) are defined, and in this case (a + b) + c = a + (b + c); (iii) for any a € E, there exists a unique element a' e £ such that a + a' = 1; (iv) if a + 1 is defined in E then a = 0. The relation < on E is defined for a, b e E by a < b if and only if there exists an element c € E such that a + c = b. This relation is a partial ordering and we write c = ba.

doi:10.1017/s1446788700016025
fatcat:irwjskea5bhmnfo4suuqrmccbi