Basic logic algebras

Abbreviation: BLA

Definition

A basic logic algebra or BL-algebra is a structure A = (A,∨,0,∧,1,·,→) such that
(A,∨,0,∧,1) is a bounded lattice,
(A,·,1) is a commutative monoid,
→ gives the residual of ·:   x·y ≤ z  ⇔   y ≤ x→z,
linearity:   ( x→y) ∨( y→x)  = 1, and
BL:   x·(x→y) = x∧y.

Remark: The BL identity implies that the lattice is distributive.

Definition

A basic logic algebra is a FLe-algebra A = (A,∨,0,∧,1,·,→) such that
linearity:   ( x→y) ∨( y→x)  = 1, and
BL:   x·(x→y) = x∧y.

Remark: The BL identity implies that the identity element 1 is the top of the lattice.

Morphisms

Let A and B be basic logic algebras. A morphism from A to B is a function h : A→B that is a homomorphism: h(x∨y) = h(x)∨h(y)  and  h(1) = 1  and  h(x∧y) = h(x)∧h(y)  and  h(0) = 0  and  h(x·y) = h(x)·h(y)  and  h(x→y) = h(x)→h(y) hold.

Some results

Examples

Properties

Finite members

Subclasses

Superclasses