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This paper deals with the sets of strict conjectures and consequences of a given collection P of premises. The set of Averaging Functions is introduced on lattices and some properties of these functions are shown. Averaging Functions allow to interpret restricted consequences as averages of premises. The subset of consequences C9*(P) and the subset of conjectures Φg*(P) defined by means of the averaging function g are introduced, and their properties are studied. This sets allow to give decomposition...
We introduce the notion of p-ideal of a QMV-algebra and we prove that the class of all p-ideals of a QMV-algebra M is in one-to-one correspondence with the class of all congruence relations of M.
We characterize d-lattices as those bounded lattices in which every maximal filter/ideal is prime, and we show that a d-lattice is complemented iff it is balanced iff all prime filters/ideals are maximal.
The concept of a basic pseudoring is introduced. It is shown that every orthomodular lattice can be converted into a basic pseudoring by using of the term operation called Sasaki projection. It is given a mutual relationship between basic algebras and basic pseudorings. There are characterized basic pseudorings which can be converted into othomodular lattices.
The concept of a 0-distributive poset is introduced. It is shown that a section semicomplemented poset is distributive if and only if it is 0-distributive. It is also proved that every pseudocomplemented poset is 0-distributive. Further, 0-distributive posets are characterized in terms of their ideal lattices.
In their recent paper on posets with a pseudocomplementation denoted by the first and the third author introduced the concept of a -ideal. This concept is in fact an extension of a similar concept introduced in distributive pseudocomplemented lattices and semilattices by several authors, see References. Now we apply this concept of a c-ideal (dually, c-filter) to complemented posets where the complementation need neither be antitone nor an involution, but still satisfies some weak conditions....
An equivalent definition of compatibility in pseudo-effect algebras is given, and its relationships with central elements are investigated. Furthermore, pseudo-MV-algebras are characterized among pseudo-effect algebras by means of compatibility.
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