## Multiplicative additive linear logic algebras

Abbreviation: MALLA

### Definition

A multiplicative additive linear logic algebra is a structure $\mathbf{A}=\langle A,\vee,\bot,\wedge,\top,+,0,\cdot,1,^\perp\rangle$ of type $\langle 2,0,2,0,2,0,1\rangle$ such that

$\langle A,\vee,\wedge,\cdot,1,^{\perp}\rangle$ is a commutative involutive residuated lattice

$\bot$ is the least element: $\bot\le x$

$\top$ is the greatest element: $x\le\top$

$+$ is the dual of $\cdot$: $x+y=(x^\perp\cdot y^\perp)^\perp$

$0$ is the dual of $1$: $0=1^\perp$

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 multiplicative additive linear logic algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism.

### Definition

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

$...$ is …: $axiom$

$...$ 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 decidable 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

[[...]] subvariety
[[...]] expansion

### Superclasses

[[...]] supervariety
[[...]] subreduct
##### Toolbox 