In this chapter, ideals of QI-algebra are considered. Given a subset of a right distributive QI-algebra, the smallest ideal containing it is constructed. Also, the notions of implicative ideal, fantastic ideal, and normal ideal in a right distributive QI-algebra are introduced, and the authors proved that these notions are equivalent.
Top2. Preliminaries
First, we recall certain definitions and properties from Saeid et al.(2017), Abbott (1967) & Chen et.al. (1995) that are required in the paper.
Definition 2.1 (Iseki, 1980) A BCI-algebra is an algebra (X,*,0) of type (2,0) satisfying the following conditions:
- (1)
(x*y)*(x*z)≤(z*y)
- (2)
x*(x*y)≤y
- (3)
x≤x
- (4)
x≤y and y≤y imply x=y
- (5)
x≤0 implies x=0
where
x≤
y is defined by
x*y=0.
If (5) is replaced by (6) 0≤x, then the algebra is called a BCK-algebra. It is known that every BCK-algebra is a BCI-algebra but not conversely. A BCK-algebra satisfying the property x*(y*x)= x for all x,y∈X is called an implicative BCK-algebra.
Several generalizations of a BCK-algebra, in the form of definitions, one can see in the paper (Saeid et al., 2017).
Definition 2.2 (Abbott, 1967) A groupoid (X,*) is called an implication algebra if it satisfies the following identities:
- (a)
(x*y)*x=x
- (b)
(x*y)*y=(y*x)*x
- (c)
X*(y*z)=y*(x*z)
for all
x,y,z∈
X.Definition 2.3 (Abbott, 1967) Let (X,*) be an implication algebra and binary operation “” on X be defined by Then is said to be a dual implication algebra. In fact, the axioms of that are as follows:
for all
x,y,z∈
X.
Chen & Oliveira (1995) proved that in any implication algebra (X,*) the identity x*y=y*y holds for all x,y∈X. We denote the identity x*x=y*y by the constant 0. The notion of BI-algebras comes from the (dual) implication algebra.