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@@ -995,7 +995,7 @@ Mika-Michalski)
 <td align="left"><i ><u >Jas Å emrl</u></i></td>
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 <a href="javascript:toggleDiv('T40')">Finite Representation Property for Relation Algebra Reducts (short talk)</a>
-<div style="display:none", id="T40"><br /><br />Membership in the Class of Representable Tarskian Relation Algebra in undecid- able. However, dropping some operations to obtain a reduct language may introduce some more favourable properties. The Finite Representation Property (FRP) is a great example of good algebraic behaviour. A signature is said to have the FRP if every rep- resentable finite structure in that signature can be represented over a finite base set. Some positive implications of the FRP for a signature include the decidability of mem- bership in the representation class for finite structures, as well as the decidability of membership in equational theory generated by the representation class. Although the property fails for the language of Relation Algebras, it remains an open question for many of its reduct languages. We examine the conjecture that a reduct language of Re- lation Algebra has the FRP if and only if it does not include negation and composition, nor meet and composition. Here we prove that any signature containing composition and negation fails to have the FRP. This preliminary result, together with FRP failing for any signature containing composition and meet [Neu16], proves the right to left implication of the conjecture. For the left to right implication, the FRP is known to hold for composition-free signatures as well as a handful of those containing composi- tion. Furthermore, we show here that in any reduct signature with composition but neither meet nor negation a [finite] representable structure has a [finite] representation if and only if it embeds into a [finite] Relation Algebra (not necessarily representable). This suggests that well known counterexamples to FRP in other signatures, like the Point Algebra will have a finite base representation in signatures conjectured to have the FRP. Finally, we look at signatures, like that of join-lattice semigroups, where embedding a finite representable structure into a finite relation algebra appears to be especially difficult and look at examples of some of the more interesting known such embeddings.
+<div style="display:none", id="T40"><br /><br />Membership in the Class of Representable Tarskian Relation Algebra is undecidable. However, dropping some operations to obtain a reduct language may introduce some more favourable properties. The Finite Representation Property (FRP) is a great example of good algebraic behaviour. A signature is said to have the FRP if every representable finite structure in that signature can be represented over a finite base set. Some positive implications of the FRP for a signature include the decidability of membership in the representation class for finite structures, as well as the decidability of membership in equational theory generated by the representation class. Although the property fails for the language of Relation Algebras, it remains an open question for many of its reduct languages. We examine the conjecture that a reduct language of Relation Algebra has the FRP if and only if it does not include negation and composition, nor meet and composition. Here we prove that any signature containing composition and negation fails to have the FRP. This preliminary result, together with FRP failing for any signature containing composition and meet [Neu16], proves the right to left implication of the conjecture. For the left to right implication, the FRP is known to hold for composition-free signatures as well as a handful of those containing composition. Furthermore, we show here that in any reduct signature with composition but neither meet nor negation a [finite] representable structure has a [finite] representation if and only if it embeds into a [finite] Relation Algebra (not necessarily representable). This suggests that well known counterexamples to FRP in other signatures, like the Point Algebra will have a finite base representation in signatures conjectured to have the FRP. Finally, we look at signatures, like that of join-lattice semigroups, where embedding a finite representable structure into a finite relation algebra appears to be especially difficult and look at examples of some of the more interesting known such embeddings.
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