Algebraic theories

Content created by Fernando Chu and Fredrik Bakke.

Created on 2023-03-20.
Last modified on 2023-04-28.

module universal-algebra.algebraic-theories where
open import foundation.dependent-pair-types
open import foundation.universe-levels

open import universal-algebra.abstract-equations-over-signatures
open import universal-algebra.signatures


An algebraic theory is a collection of abstract equations over a signature S that we consider to 'hold' in the theory. It is algebraic in the sense that we only require equations involving function symbols from the signature, in contrast to, say, requiring additional types of relations.



module _
  {l1 : Level} (Sg : signature l1)

  Theory : (l2 : Level)  UU (l1  lsuc l2)
  Theory l2 = Σ (UU l2)  B  (B  Abstract-Equation Sg))

  index-Theory : {l2 : Level}  Theory l2  UU l2
  index-Theory = pr1

  index-Abstract-Equation-Theory :
    { l2 : Level}
    ( Th : Theory l2) 
    ( index-Theory Th) 
    Abstract-Equation Sg
  index-Abstract-Equation-Theory Th e = pr2 Th e

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