Radical of filters on equality algebra

In this paper, by using the concept of maximal filter of equality algebra, we introduce radical of equality algebra. Then some equivalence definitions of it and some related properties are investigated. Then by using this notion, we introduce the concept of semi-maximal filter and prime-like filter on equality algebras and the relation between them and other filters of equality algebra are investigated. Finally, by using the notion of prime-like filters, we introduce a topology on equality algebra.

Dual annihilator filters of commutative BE-algebras

Some properties of dual annihilator filters of commutative [Formula: see text]-algebras are studied. It is proved that the class of all dual annihilator filters of a BE-algebra is a complete Boolean algebra. A set of equivalent conditions is derived for every prime filter of a commutative [Formula: see text]-algebra to become a maximal filter.

States on bounded commutative residuated lattices

We define states on bounded commutative residuated lattices and consider their property. We show that, for a bounded commutative residuated lattice X, (1)If s is a state, then X/ker(s) is an MV-algebra.(2)If s is a state-morphism, then X/ker(s) is a linearly ordered locally finite MV-algebra.Moreover we show that for a state s on X, the following statements are equivalent: (i)s is a state-morphism on X.(ii)ker(s) is a maximal filter of X.(iii)s is extremal on X.

Filters ofR0-algebras

The notion of fated filters inR0-algebras is introduced. Characterizations of (fated) filters are given. A filter generated by a set is established. By introducing the notion of finite⊙-property, we show that ifFis a nonempty subset of anR0-algebraLthat has the finite⊙-property, then there exists a maximal filter ofLcontainingF.