## Linear logic algebras

Abbreviation: LLA

### Definition

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

$\langle A,\vee,\wedge,\cdot,1,\neg\rangle$ is an involutive residuated lattice

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

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

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

$0$ is the dual of $1$: $0=\neg 1$

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 … . A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x \ldots y)=h(x) \ldots h(y)$

### 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 (value, see description) 1)

### 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

### References

1) F. Lastname, \emph{Title}, Journal, \textbf{1}, 23–45 MRreview