Abstract.
Weak effect algebras are based on a commutative, associative and cancellative partial addition; they are moreover endowed with a partial order which is compatible with the addition, but in general not determined by it. Every BL-algebra, i.e. the Lindenbaum algebra of a theory of Basic Logic, gives rise to a weak effect algebra; to this end, the monoidal operation is restricted to a partial cancellative operation.
We examine in this paper BL-effect algebras, a subclass of the weak effect algebras which properly contains all weak effect algebras arising from BL-algebras. We describe the structure of BL-effect algebras in detail. We thus generalise the well-known structure theory of BL-algebras.
Namely, we show that BL-effect algebras are subdirect products of linearly ordered ones and that linearly ordered BL-effect algebras are ordinal sums of generalised effect algebras. The latter are representable by means of linearly ordered groups.
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Received October 31, 2006; accepted in final form February 6, 2007.
This research was partially supported by the German Science Foundation (DFG) as part of the Collaborative Research Center “Computational Intelligence” (SFB 531).
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Vetterlein, T. Weak effect algebras. Algebra univers. 58, 129–143 (2008). https://doi.org/10.1007/s00012-008-2052-6
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DOI: https://doi.org/10.1007/s00012-008-2052-6