Lattice effect algebras generalize orthomodular lattices and -algebras. We describe all complete modular atomic effect algebras. This allows us to prove the existence of ordercontinuous subadditive states (probabilities) on them. For the separable noncomplete ones we show that the existence of a faithful probability is equivalent to the condition that their MacNeille complete modular effect algebra.
We prove that every modular function on a multilattice with values in a topological Abelian group generates a uniformity on which makes the multilattice operations uniformly continuous with respect to the exponential uniformity on the power set of .
Let be an algebraic structure of type and a set of open formulas of the first order language . The set of all subsets of closed under forms the so called lattice of -closed subsets of . We prove various sufficient conditions under which the lattice is modular or distributive.
In a Boolean Algebra B, an inequality f(x,x --> y)) ≤ y satisfying the condition f(1,1)=1, is considered for defining operations a --> b among the elements of B. These operations are called Conditionals'' for f. In this paper, we obtain all the boolean Conditionals and Internal Conditionals, and some of their properties as, for example, monotonicity are briefly discussed.
We show that an ideal I of an MV-algebra A is linearly ordered if and only if every non-zero element of I is a molecule. The set of molecules of A is contained in Inf(A) ∪ B2(A) where B2(A) is the set of all elements x ∈ A such that 2x is idempotent. It is shown that I ≠ {0} is weakly essential if and only if B⊥ ⊂ B(A). Connections are shown among the classes of ideals that have various combinations of the properties of being implicative, essential, weakly essential, maximal or prime.
The concept of monadic MV-algebra was recently introduced by A. Di Nola and R. Grigolia as an algebraic formalization of the many-valued predicate calculus described formerly by J. D. Rutledge [9]. This was also genaralized by J. Rachůnek and F. Švrček for commutative residuated -monoids since MV-algebras form a particular case of this structure. Basic algebras serve as a tool for the investigations of much more wide class of non-classical logics (including MV-algebras, orthomodular lattices and...