polyadic algebra with equality
Let be a polyadic algebra. An equality predicate on is a function such that
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1.
for any and any
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2.
for every , and
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, where , and denotes the function that maps to , and constant everywhere else.
Heuristically, we can interpret the conditions above as follows:
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1.
if and if we replace by, say , and by , then .
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2.
for every variable
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3.
if we have a propositional function that is true, and , then the proposition obtained from by replacing all occurrences of by is also true.
The second condition is also known as the reflexive property of the equality predicate , and the third is known as the substitutive property of
A polyadic algebra with equality is a pair where is a polyadic algebra and is an equality predicate on . Paul Halmos introduced this concept and called this simply an equality algebra.
Below are some basic properties of the equality predicate in an equality algebra :
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(symmetric property)
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, where in the is the transposition on that swaps and and leaves everything else fixed.
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if a variable is not in the support of , then .
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for all and all whenever .
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for all where .
Remarks
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The degree and local finiteness of a polyadic algebra are defined as the degree and the local finiteness and degree of its underlying polyadic algebra .
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It can be shown that every locally finite polyadic algebra of infinite degree can be embedded (as a polyadic subalgebra) in a locally finite polyadic algebra with equality of infinite degree.
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Like cylindric algebras, polyadic algebras with equality is an attempt at “converting” a first order logic (with equality) into algebraic form, so that the logic can be studied using algebraic means.
References
- 1 P. Halmos, Algebraic Logic, Chelsea Publishing Co. New York (1962).
- 2 B. Plotkin, Universal Algebra, Algebraic Logic, and Databases, Kluwer Academic Publishers (1994).
Title | polyadic algebra with equality |
Canonical name | PolyadicAlgebraWithEquality |
Date of creation | 2013-03-22 17:51:37 |
Last modified on | 2013-03-22 17:51:37 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 10 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 03G15 |
Synonym | equality algebra |
Related topic | CylindricAlgebra |
Defines | equality predicate |
Defines | substitutive |
Defines | reflexive |
Defines | symmetric |
Defines | transitive |