monoidal_t-norm_logic_algebras

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=====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 $\mathbf{A}=\langle A,...\rangle$ of type $\langle
...\rangle$ such that

$...$ is ...:  $axiom$
  
$...$ is ...:  $axiom$

====Examples====
Example 1: 

====Basic results====


====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  |
^[[Equational theory]]                | |
^[[Quasiequational theory]]           | |
^[[First-order theory]]               | |
^[[Locally finite]]                   |no |
^[[Residual size]]                    |unbounded |
^[[Congruence distributive]]          |yes |
^[[Congruence modular]]               |yes |
^[[Congruence $n$-permutable]]        |yes, $n=2$ |
^[[Congruence regular]]               |no |
^[[Congruence uniform]]               |no |
^[[Congruence extension property]]    | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]      | |
^[[Amalgamation property]]            | |
^[[Strong amalgamation property]]     | |
^[[Epimorphisms are surjective]]      | |

====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====

[(Ln19xx>
F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]] 
)]

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