Hypercodes, right convex languages and their syntactic monoids

Abstract:

If ${X^ * }$ is the free monoid generated by the alphabet $X$, then any subset $L$ of ${X^ * }$ is called a language over $X$. If ${P_L}$ is the principal congruence determined by $L$, then the quotient monoid ${\text {syn}}(L) = {X^ * }/{P_L}$ is called the syntactic monoid of $L$. A hypercode over $X$ is any set of nonemtpy words that are noncomparable with respect to the embedding order of ${X^ * }$. If $H$ is a hypercode, then the language $\tilde H = \{ x|x \in {X^ * }$ and $a \leqslant x$ for some $a \in H\}$ is a right convex ideal of ${X^ * }$. The syntactic monoid ${\text {syn}}(\tilde H)$ can be characterized as a monoid with a disjunctive $\mu$-zero. The two particular interesting cases when ${\text {syn}}(\tilde H)$ is a nil monoid and when ${\text {syn}}(\tilde H)$ is a semillatice are also characterized.