Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:   show that relativized versions of relation set algebras and cylindric set alge- bras have undecidable equational theories if  include coordinatewise versions of the counting operations into the similarity type.  apply these results to the guarded fragment of first-order logic. ?1. Introduction. Relativized algebras of relations are extensively investigated in the literature, cf., e.g., (HMT, HMTAN, Ma82, Mo93, Ne9 1). In general, relativized versions of algebras of relations have a nicer behavior from the computational point of view than the original versions. ,  concentrate on (un)decidability.  show that if  include coordinatewise versions of the counting operations into the similarity type, then the expressive por is strong enough to interpret the tiling problem into the equational theories of relativized relation set algebras and cylindric-relativized set algebras of dimension (at least) three. Thus these equational theories must be undecidable. Finally, in the last section,  apply these results to logic: the corresponding versions of the guarded fragment of first-order logic and of arrow logic are unde- cidable. Acknowledgments: Thanks are due to Hajnal Andreka and an anonymous referee for careful reading and valuable suggestions. Special thanks are due to the members of the Group of Algebraic Logicians in London: Robin Hirsch, Ian Hodkinson and Mark Reynolds. 1.1. Relativization. Relativization of an algebra amounts to intersecting all its elements with a fixed set (usually an element of the algebra or a subset of the unit) and to defining the operations using this set as the unit of the new algebra. It turned out that if  relativize (set) algebras of relations with arbitrary, sym- metric and/or reflexive elements, then  get a class of algebras with nice algebraic properties. For instance, while relation (set) algebras and cylindric (set) algebras of dimension at least three have undecidable equational theories, the sets of equations valid in the above relativizations are decidable. Traditionally, during relativization  keep the original similarity type in the case of relation algebras: Booleans, composition, converse, identity. As a consequence, some operations that are definable in the original version are not available after relativization. An example is the global counting operations once
['Szabolcs Mikulás', 'Maarten Marx']
I am writing a research paper on a idea UNDECIDABLE RELATIVIZATIONS OF ALGEBRAS OF RELATIONS
[{"from": "human", "value": "I am writing a research paper on a idea UNDECIDABLE RELATIVIZATIONS OF ALGEBRAS OF RELATIONS"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:   show that relativized versions of relation set algebras and cylindric set alge- bras have undecidable equational theories if  include coordinatewise versions of the counting operations into the similarity type.  apply these results to the guarded fragment of first-order logic. ?1. Introduction. Relativized algebras of relations are extensively investigated in the literature, cf., e.g., (HMT, HMTAN, Ma82, Mo93, Ne9 1). In general, relativized versions of algebras of relations have a nicer behavior from the computational point of view than the original versions. ,  concentrate on (un)decidability.  show that if  include coordinatewise versions of the counting operations into the similarity type, then the expressive por is strong enough to interpret the tiling problem into the equational theories of relativized relation set algebras and cylindric-relativized set algebras of dimension (at least) three. Thus these equational theories must be undecidable. Finally, in the last section,  apply these results to logic: the corresponding versions of the guarded fragment of first-order logic and of arrow logic are unde- cidable. Acknowledgments: Thanks are due to Hajnal Andreka and an anonymous referee for careful reading and valuable suggestions. Special thanks are due to the members of the Group of Algebraic Logicians in London: Robin Hirsch, Ian Hodkinson and Mark Reynolds. 1.1. Relativization. Relativization of an algebra amounts to intersecting all its elements with a fixed set (usually an element of the algebra or a subset of the unit) and to defining the operations using this set as the unit of the new algebra. It turned out that if  relativize (set) algebras of relations with arbitrary, sym- metric and/or reflexive elements, then  get a class of algebras with nice algebraic properties. For instance, while relation (set) algebras and cylindric (set) algebras of dimension at least three have undecidable equational theories, the sets of equations valid in the above relativizations are decidable. Traditionally, during relativization  keep the original similarity type in the case of relation algebras: Booleans, composition, converse, identity. As a consequence, some operations that are definable in the original version are not available after relativization. An example is the global counting operations once"}]