# Algebra Seminar talk

2005-11-18

Zdenka **Riecanova***Some families of effect algebras*

Abstract:

Lattice ordered effect algebras generalize orthomodular
lattices (which may include noncompatible pairs of elements) and
MV-algebras (which may include unsharp elements).

Thus effect algebras may include unsharp elements and noncompatible pairs of elements as well. On the other hand, for every effect algebra E, the set S(E) of all sharp elemnts of E is an orthomodular lattice and every maximal subset M of pairwise compatible elements of E is an MV-algebra called a block of E. Further the intersection of all blocks of E (denoted B(E)) with the set S(E) is a Boolean algebra C(E) called a center of E.

In general, S(E), B(E), C(E) and blocks of E are different sub-effect algebras and full sublattices of the lattice effect algebra E. Their special properties may define a special families of effect algebras whose algebraic structures can be described. Consequently, questions about existence of states, (o)-continuous states and probabilities on them, or properties of direct product decompositions, completions and some others can be answered.