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Basic logic algebras

Abbreviation: BLA

Definition

A basic logic algebra or BL-algebra is a structure $\mathbf{A}=\langle A,\vee ,0,\wedge ,1,\cdot ,arrow \rangle $ such that

$\langle A,\vee ,0,\wedge ,1\rangle $ is a bounded lattice

$\langle A,\cdot ,1\rangle $ is a commutative monoid

$arrow $ gives the residual of $\cdot $: $x\cdot y\leq z\Longleftrightarrow y\leq xarrow z$

prelinearity: $( xarrow y) \vee ( yarrow x) =1$

BL: $x\cdot(xarrow y)=x\wedge y$

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

Definition

A basic logic algebra is a FLe-algebra $\mathbf{A}=\langle A,\vee ,0,\wedge ,1,\cdot ,arrow \rangle $ such that

linearity: $( xarrow y) \vee ( yarrow x) =1$

BL: $x\cdot (xarrow y)=x\wedge y$

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

Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be basic logic algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:Aarrow B$ that is a homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(1)=1$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(xarrow y)=h(x)arrow h(y)$

Examples

Example 1:

Basic results

Properties

Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &2\\ f(4)= &5\\ \end{array}$

The number of subdirectly irreducible BL-algebras of size $n$ is $2^{n-2}$.

\hyperbaseurl{http://math.chapman.edu/structures/files/}

Subclasses

Superclasses

References