Linear logic algebras

Abbreviation: LLA


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 $1$: $0=\neg 1$

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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.


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: $h(x \ldots y)=h(x) \ldots h(y)$


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:

Basic results


Finite members


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)= &\\



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


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


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

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