# Algebraic theories

Content created by Fernando Chu and Fredrik Bakke.

Created on 2023-03-20.

module universal-algebra.algebraic-theories where

Imports
open import foundation.dependent-pair-types
open import foundation.universe-levels

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


## Idea

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.

## Definitions

### Theories

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

Theory : (l2 : Level) → UU (l1 ⊔ lsuc l2)
Theory l2 = Σ (UU l2) (λ 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