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}: $(x\to y)\vee (y\to x)=1$

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 …: $axiom$

Example 1:

Properties

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

Classtype variety no unbounded yes yes yes, $n=2$ no no

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