Question

BOURBAKI, inside his book on ALGEBRA defines and provides explicit constructions concerning the concepts of free magma, free monoid (and implicitele free semi-group) and free group, and as well free commutative monoid (and implicitely free commutative semi-group) and free commutative group over a set X; It seems clear that the concept of free commutative magma also makes sense, but doescanyone know about an explicit construction for the free commutative magma over the set X ? Gérard LANG

Answer 1

The free magma on $X$ consists of finite sequences of length $1$ or $2$, which consist of finite sequences of length $1$ or $2$, etc., of elements of $X$; the magma operation is just concatenation, i.e. $mn = (m,n)$. For example, $(a,((b,c),d))$ is such a sequence, where $a,b,c,d$ are in $X$. Elements may be visualized as finite binary trees, whose leaves are labelled in $X$. The examples gives:

To give a formal construction, define by recursion the sets $X_n$ of elements of height $n$ by $X_0 = X$ and $X_{n+1} = X_n + X_n^2$ (disjoint union). Then the disjoint union of $X_n$ is the free magma on $X$.

Now the free commutative magma is obtained by taking the quotient with respect to the smallest congruence relation satisfying $(a,b) \sim (b,a)$. This can be visualized with trees: Every branch can be rotated freely. So for example, now we don't distinguish between the trees

.

Note, however, that this does not justify that we may replace every bracket $(...)$ with $\{...\}$. Namely, $(a,a)$, which has two leaves, has to be distinguished from $(a)$, which has only one.

Answer 2

As a set-theorist, I'd view the free commutative magma $M$ on $X$ as the family of hereditarily 1-or-2-element sets over $X$, together with $X$ itself. Here the members of $X$ are to be regarded as atoms (= urelements), not as sets.

In more detail, $M$ consists of the members of $X$ together with all those sets $a$ for which there is a transitive set $t$ ("transitive" means that, if a set $s$ is a member of $t$ then so are all members of $s$) such that $a\in t$ and such that every member of $t$ is either a member of $X$ or a set of cardinality 1 or 2.