notes_19_02_08 - Applied Logic Lecture Notes 1 Equational...

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Applied Logic Lecture Notes 19-02-08 1 Equational Logic A signature is a map σ : F σ → { 0 , 1 , 2 , } where F σ is a family of distinct symbols. The terms of signature σ , or simply σ -terms , are defined by the following rules: Every variable symbol x 0 , x 1 , is a σ -term. 1 If f ∈ F σ and t 1 , , t σ ( f ) are σ -terms, then f t 1 t σ ( f ) is also a σ -term. Note that if σ ( f ) = 0 , then f is a σ -term. A ground σ -term is one that doesn’t involve any variable symbols. A σ -equation is an expression of the form t u where t, u are σ - terms. Example. The theory of groups has signature { e 0 , square 1 1 , square · square 2 } and equations x · ( y · z ) ( x · y ) · z x · e x x · x 1 e e · x x x 1 · x e The theory of Abelian groups is obtained by adding the equation x · y y · x . Example. The theory of rings has signature { 0 0 , 1 0 , square 1 , square + square 2 , square × square 2 } and equations x + ( y + z ) ( x + y ) + z x × ( y × z ) ( x × y ) × z 0 + x x 1 × x x x + 0 x x × 1 x x + ( x ) 0 ( x + y ) × z ( x × z ) + ( y × z ) ( x ) + x 0 x × ( y + z ) ( x × y ) + ( x × z ) Example. The theory of Boolean algebras has signature { 0 0 , 1 0 , square 1 , square square 2 , square square 2 } and equations x y y x x y y x x ( y z ) ( x y ) z x ( y z ) ( x y ) z x ( y z ) ( x y ) ( x z ) x ( y z ) ( x y ) ( x z ) x ( x y ) x x ( x y ) x x x 0 x x 1 x 0 0 x 1 1 x 1 x x 0 x 1. We implicitly assume that the set of variable symbols and function symbols are disjoint.
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