Linear logic algebras

Abbreviation: LLA

Definition

A 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 $1$: $0=\neg 1$

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

Definition

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

$...$ is …: $axiom$

$...$ 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

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

Superclasses

[[...]] supervariety
[[...]] subreduct

References


1) F. Lastname, Title, Journal, 1, 23–45 MRreview