USC Talk

<h2 align=center>Decidability of some varieties of residuated lattices</h2>
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<table cellspacing=3 border=3 cellpadding=3>
<tr>
<td>Variety</td>
<td>Equations</td>
<td>Equational Theory</td>
<td>Universal Theory</td>
</tr>

<tr>
<td>Residuated Lattices</td>
<td> </td>
<td>Decidable Okada & Terui 99</td>
<td>Undecidable</td>
</tr>

<tr>
<td>Commutative</td>
<td>`x.y=y.x`</td>
<td>Decidable Okada & Terui 99</td>
<td> </td>
</tr>

<tr>
<td>Distributive</td>
<td>`x^^(yvvz)`= `(x^^y)vv(x^^z)`</td>
<td> </td>
<td>Undecidable Galatos 00</td>
</tr>

<tr>
<td>Comm. & Distr.</td>
<td> </td>
<td> </td>
<td>Undecidable Galatos 00</td>
</tr>

<tr>
<td>V(Resid. Chains)=RLC</td>
<td> </td>
<td> </td>
<td> </td>
</tr>

<tr>
<td>Commutative RLC</td>
<td> </td>
<td> </td>
<td> </td>
</tr>

<tr>
<td>l-groups=LG</td>
<td>`x.(e/x)=e`</td>
<td>Decidable Holland & McCleary 79</td>
<td>Undecidable Glass & Gurevich 83</td>
</tr>

<tr>
<td>`bf(LG)^-`</td>
<td> </td>
<td>Decidable J. 00</td>
<td> </td>
</tr>

<tr>
<td>Abelian l-groups</td>
<td> </td>
<td>Decidable</td>
<td>Decidable Khisamiev 66</td>
</tr>

<tr>
<td>`bfV(bfZ^-)`</td>
<td> </td>
<td>Decidable</td>
<td> </td>
</tr>

<tr>
<td>Integral</td>
<td>`x<=e`</td>
<td>Decidable Okada & Terui 99</td>
<td> </td>
</tr>

<tr>
<td>Comm. Integral</td>
<td> </td>
<td>Decidable Okada & Terui 99</td>
<td>Decidable Blok & van Alten 01</td>
</tr>

<tr>
<td>Brouwerian Algebras</td>
<td>`x.y=x^^y`</td>
<td>Decidable</td>
<td>Decidable McKinsey & Tarski 44</td>
</tr>

<tr>
<td>Integ. Comm. & n-potent</td>
<td>`x^n=x^(n+1)`</td>
<td>Decidable</td>
<td>Decidable Blok & van Alten 01</td>
</tr>

<tr>
<td>Modular RL</td>
<td> </td>
<td>Undecidable J. 01</td>
<td>Undecidable</td>
</tr>

<tr>
<td>Cancellative RL</td>
<td>`x.y/y=x=y\y.x`</td>
<td> </td>
<td> </td>
</tr>

</table>
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