L-effect Algebras

Abstract

L-effect algebras are introduced as a class of L-algebras which specialize to all known generalizations of effect algebras with a $$\wedge $$ -semilattice structure. Moreover, L-effect algebras X arise in connection with quantum sets and Frobenius algebras. The translates of X in the self-similar closure S(X) form a covering, and the structure of X is shown to be equivalent to the compatibility of overlapping translates. A second characterization represents an L-effect algebra in the spirit of closed categories. As an application, it is proved that every lattice effect algebra is an interval in a right $$\ell $$ -group, the structure group of the corresponding L-algebra. A block theory for generalized lattice effect algebras, and the existence of a generalized OML as the subalgebra of sharp elements are derived from this description.