Monoid Factorization

A monoid factorization of a Free Monoid A ∗ superscript 𝐴 A^{*} typically defined as a sequence of subsets of words X i ⊆ A ∗ subscript 𝑋 𝑖 superscript 𝐴 X_{i}\subseteq A^{*} indexed by a totally ordered set ( I , ≤ ) 𝐼 (I,\leq) such that every word w : A ∗ : 𝑤 superscript 𝐴 w:A^{*} can be written as w = x i 1 ⁢ x i 2 ⁢ … ⁢ x i n 𝑤 subscript 𝑥 subscript 𝑖 1 subscript 𝑥 subscript 𝑖 2 … subscript 𝑥 subscript 𝑖 𝑛 w=x_{i_{1}}\,x_{i_{2}}\,\ldots\,x_{i_{n}} with i 1 ≤ i 2 ≤ … ≤ i n subscript 𝑖 1 subscript 𝑖 2 … subscript 𝑖 𝑛 i_{1}\leq i_{2}\leq\ldots\leq i_{n} .

A more elegant definition is to regard ( I , ≤ ) 𝐼 (I,\leq) as a category, and consider a diagram of monoids M : I → Mon : 𝑀 → 𝐼 Mon M:I\to\mathrm{Mon} . From this perspective, a factorization of A ∗ superscript 𝐴 A^{*} is a diagram of monoids such that A ∗ superscript 𝐴 A^{*} is a Colimit.