Lecture Notes on Bayesian Estimation and Classification
《Lecture Notes on Bayesian Estimation and Classification》 · 第 60 页 · Section 2.3「Varieties of Groups」，小节「Free $\mathfrak{V}$-Groups」
Let $\mathfrak{V}$ be a variety, $F$ a group in $\mathfrak{V}$, $X$ a nonempty set, and $\sigma:X\to F$ a function. Then $(F,\sigma)$, or simply $F$, is $\mathfrak{V}$-free on $X$ if for each function $\alpha$ from $X$ to a $\mathfrak{V}$-group $G$ there exists a unique homomorphism $\beta:F\to G$ such that $\sigma\beta=\alpha$. When $\mathfrak{V}$ is the variety of all groups, this is just the definition of a free group on $X$. A group which is $\mathfrak{V}$-free on some set is called a free $\mathfrak{V}$-group. Groups which are free in some variety are often said to be relatively free.