Provable formulas

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Other provable formules are given by [[List of isomorphisms|isomorphisms]].

Revision as of 21:09, 25 April 2013

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In many of the cases below the converse implication does not hold.

Contents

Distributivities

A\plus (B\with C) \limp (A\plus B)\with (A\plus C)

A\tens (B\parr C) \limp (A\tens B)\parr C

Factorizations

(A\with B)\plus (A\with C) \limp A\with (B\plus C)

Additive structure


\begin{array}{rcl}
  A\with B \limp A &\quad& A\with B \limp B\\
  (C\limp A)\with(C\limp B) &\limp& C\limp(A\with B)\\
  A &\limp& A\with A\\
  A \limp A\plus B &\quad& B \limp A\plus B\\
  (A\limp C)\with(B\limp C) &\limp& (A\plus B)\limp C\\
  A\plus A &\limp& A\\
\end{array}

Exponential structure

Provable formulas involving exponential connectives only provide us with the lattice of exponential modalities.


\begin{array}{rclcrcl}
  \oc A &\limp& A &\quad& A&\limp&\wn A\\
  \oc A &\limp& 1 &\quad& \bot &\limp& \wn A\\
  \oc A &\limp& \oc A\tens\oc A &\quad& 
  \wn A\parr\wn A &\limp& \wn A\\
  \oc A &\limp& \oc\oc A &\quad& \wn\wn A &\limp& \wn A\\
\end{array}

Monoidality of exponential


\begin{array}{rclcrcl}
  \oc A\tens\oc B &\limp& \oc(A\tens B) &\quad&
  \one &\limp& \oc\one\\
\end{array}


Other provable formules are given by isomorphisms.

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