In mathematics, BCI and BCK algebras are

algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...

s in

universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular Group (mathematics), groups as ...

, which were introduced by Y. Imai, K. Iséki and S. Tanaka in 1966, that describe fragments of the propositional calculus involving implication known as BCI and BCK logics.

Definition

BCI algebra

An algebra (in the sense of universal algebra)

\left( X;\ast
,0\right)

of type

\left( 2,0\right)

is called a ''BCI-algebra'' if, for any

x,y,z\in X

, it satisfies the following conditions. (Informally, we may read

0

as "truth" and

x\ast y

as "

y

implies

x

".) ; BCI-1:

\left( \left( x\ast y\right) \ast \left( x\ast z\right)
\right) \ast \left( z\ast y\right) =0

; BCI-2:

\left( x\ast \left( x\ast y\right) \right) \ast y=0

; BCI-3:

x\ast x=0

; BCI-4:

x\ast y=0 \land y\ast x=0\implies x=y

; BCI-5:

x\ast 0=0 \implies x=0

BCK algebra

A BCI-algebra

\left( X;\ast ,0\right)

is called a ''BCK-algebra'' if it satisfies the following condition: ; BCK-1:

\forall x\in X: 0\ast x=0.

A partial order can then be defined as ''x'' ≤ ''y'' iff x * y = 0. A BCK-algebra is said to be ''commutative'' if it satisfies: :

x\ast (x\ast y)= y\ast (y\ast x)

In a commutative BCK-algebra ''x'' * (''x'' * ''y'') = ''x'' ∧ ''y'' is the

greatest lower bound In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...

of ''x'' and ''y'' under the partial order ≤. A BCK-algebra is said to be bounded if it has a largest element, usually denoted by 1. In a bounded commutative BCK-algebra the least upper bound of two elements satisfies ''x'' ∨ ''y'' = 1 * ((1 * ''x'') ∧ (1 * ''y'')); that makes it a

distributive lattice In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set ...

Examples

Every

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...

is a BCI-algebra, with * defined as group subtraction and 0 defined as the group identity. The subsets of a set form a BCK-algebra, where A*B is the difference A\B (the elements in A but not in B), and 0 is the

empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in oth ...

. A

Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas ...

is a BCK algebra if ''A''*''B'' is defined to be ''A''∧¬''B'' (''A'' does not imply ''B''). The bounded commutative BCK-algebras are precisely the

MV-algebra In abstract algebra, a branch of pure mathematics, an MV-algebra is an algebraic structure with a binary operation \oplus, a unary operation \neg, and the constant 0, satisfying certain axioms. MV-algebras are the algebraic semantics of Łukasie ...

References

* * * * * * * Y. Huang, ''BCI-algebra'', Science Press, Beijing, 2006. * *{{citation, first=K. , last=Iséki, title=An algebra related with a propositional calculus, journal= Proc. Japan Acad. Ser. A Math. Sci. , volume= 42 , year=1966, pages= 26–29, url=http://projecteuclid.org/euclid.pja/1195522171, doi=10.3792/pja/1195522171, doi-access=free Algebraic structures Universal algebra