Abstract
In any logical algebraic structures, by using of different kinds of filters, one can construct various kinds of other logical algebraic structures. With this inspirations, in this paper by considering a hoop algebra or a hoop, that is introduced by Bosbach, the notion of co-filter on hoops is introduced and related properties are investigated. Then by using of co-filter, a congruence relation on hoops is defined, and the associated quotient structure is studied. Thus Brouwerian semilattices, Heyting algebras, Wajsberg hoops, Hilbert algebras and BL-algebras are obtained.
Highlights
Non-classical logics are formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic
We investigate under which conditions the quotient structure of this congruence relation will be Brouwerian semilattice, Heyting algebra, Wajsberg hoop, Hilbert algebra and BL-algebra
As you see we investigated the relation among the quotient hoop AI that is made by a co-filter I with other logical algebras such as Brouwerian semi-lattice, Heyting algebra, Hilbert algebra, Wajsberg hoop and BL-algebra
Summary
Non-classical logics (or called alternative logics) are formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic. Bosbach [1,2] proposed the concept of hoop which is a nice algebraic structure to research the many-valued logical system whose propositional value is given in a lattice. For various information on hoops, refer to [3,4,5,6,7,8]. We introduce the notion of co-filter in hoops and we get some properties of it. We construct a congruence relation by using co-filters on hoops. We investigate under which conditions the quotient structure of this congruence relation will be Brouwerian semilattice, Heyting algebra, Wajsberg hoop, Hilbert algebra and BL-algebra
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