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I can't give a proof of this fact: "Given a non-empty set $X$ and an equational class $V$, then $V$ contains a free object on $X$." $V$ is a class of algebras of a certain language of algebras $L$ and $V$, qua equational class, is $M(\Sigma)$ where $\Sigma$ is a set of identities and $M(\Sigma))$ indicates the class of $L$-algebras satisfying $\Sigma$. I wanna show that the couple $((W_{X},\Sigma),F)$ is an $L$-algebra, where $W_{X}$ is the set of words in the alphabet $X$ and with the only relations between words are given by $\Sigma$ and $F$ is the set of operations on $W_{X}$ given by the language $L$. The problem is: how to define $F$? Is this the right way to prove this fact? I don't know much about cathegory theory so I am trying this way.

SilvioM
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  • $F$ should be defined beforehand. Otherwise, how do you define the equations that determine $V$? – amrsa Mar 29 '21 at 13:05
  • Ok now maybe it's more clear. Let X a non-empty set and $L$ a language of algebras and $L_{i}$ the function symbols in $L$ of arity i. I define the set of terms of X, T(X) as the smallest set such that:
    1. $X \cup L_{0} \subset T(X)$ ;

    2)If $p_{1},...,p_{n} \in T(X)$ and $f \in L_{n}$ then $f(p_{1},...,p_{n}) \in T(X)$. Now I define an identity of type $L$ on $X$ as an ordered pair $(p,q)$, where $p,q \in T(X)$. Now the free object on X in the language $L$ is the pair $(W,F)$ where $W$ is the set of words on T(X) with relations given by $\Sigma$ and operations are "just strings".

     – SilvioM Mar 29 '21 at 15:03
  • I continue here. Operations $F$ are just the eveluation in $W$ of function symbols in $L$. Does this have sense? The way in which opertions work depends only from $\Sigma$, am I right? – SilvioM Mar 29 '21 at 15:07
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    Well, it seems that you are skipping one step: you have to take the quotient of $T(X)$ by the congruence generated by the identities satisfied in $V$. Otherwise you just get the free object of the given type, but perhaps the variety $V$ is only a sub variety of that, I mean, if it satisfies some identities. For example, if the type consists of one binary operation, and $V$ is the variety of semigroups, then you have to take the quotient of $T(X)$ by $\Theta(x(yz),(xy)z)$. – amrsa Mar 29 '21 at 15:13
  • Oh, I see. Thank you so much! – SilvioM Mar 29 '21 at 15:16
  • Check, for example, this book of Universal Algebra, Definition II.10.9 – amrsa Mar 29 '21 at 15:16

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