Monoidal t-norm logic algebras

Abbreviation: MTLA

Definition

A \emph{monoidal t-norm logic algebra} is a FLew-algebra $\mathbf{A}=\langle A, \vee, \wedge, \cdot, 1, \to, 0\rangle$ such that

$\cdot$ is \emph{prelinear}: $(x\to y)\vee (y\to x)=1$

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It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.

Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be monoidal t-norm logic algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x \vee y)=h(x) \vee h(y)$, $h(x \wedge y)=h(x) \wedge h(y)$, $h(x \cdot y)=h(x) \cdot h(y)$, $h(x \to y)=h(x) \to h(y)$, $h(0)=0$

Definition

An \emph{…} is a structure $\mathbf{A}=\langle A,\ldots\rangle$ of type $\langle …\rangle$ such that

$\ldots$ is …: $axiom$

$\ldots$ is …: $axiom$

Examples

Example 1:

Basic results

Properties

Finite members

$\begin{array}{lr}

f(1)= &1\\
f(2)= &\\
f(3)= &\\
f(4)= &\\
f(5)= &\\

\end{array}$ $\begin{array}{lr}

f(6)= &\\
f(7)= &\\
f(8)= &\\
f(9)= &\\
f(10)= &\\

\end{array}$

Subclasses

[[Basic logic algebras]]

Superclasses

[[Representable FL$_w$ algebras]]
[[Representable FL$_e$ algebras]]
[[Distributive FL$_{ew]]$ algebras}
[[Representable commutative integral residuated lattices]] reduced type

References


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