Weak effect algebras were introduced by the author as a generalization of effect algebras and pseudoeffect algebras. It was shown that having a basic algebra, we can restrict its binary operation to orthogonal elements only and what we get is just a weak effect algebra. However, the converse construction is impossible due to the fact that the underlying poset of a basic algebra is a lattice which need not be true for weak effect algebras. Hence, we found a weaker structure than a basic algebra which can serve as a representation of a weak effect algebra.
It is shown that every effect algebra with a full set of states can be represented as a so-called numerical algebra introduced in the paper. For numerical algebras there are introduced tense operators which indicate dynamical changes of quantum events depending on variability of states. These operators enable to recognize an effect algebra with a full set of states as a temporal logic where events are quantified by these tense operators. The problem of representation of tense operators on a given numerical algebra is solved.
It was shown by the author and R. Halaš that every effect algebra can be organized into a conditionally residuated structure. Skew effect algebras were introduced as a non-associative modification of effect algebras. Hence, there is natural question if a similar characterization by a certain residuated structure is possible. For this we use the so-called skew residuated structure introduced recently by the author and J. Krňávek. It is shown that this is really a suitable tool for the representation.
We introduce two unary operators G and H on a relatively pseudocomplemented lattice which form an algebraic axiomatization of the tense quantifiers “it is always going to be the case that” and “it has always been the case that”. Their axiomatization is an extended version for the classical logic and it is in accordance with these operators on many-valued Łukasiewicz logic. Finally, we get a general construction of these tense operators on complete relatively pseudocomplemented lattice which is a power lattice via the so-called frame.