On the realisation of monoids as syntactic monoids of languages Let $L \subseteq X^{\ast}$ be some language, then we define the syntactic congruence as
$$
 u \sim v :\Leftrightarrow \forall x, y\in X^{\ast} : xuy \in L \leftrightarrow xvy \in L
$$
and the quotient monoid $X^{\ast} / \sim_L$ is called the syntactic monoid of $L$.


  Now what monoids arise as syntactic monoids of languages? I found languages for symmetric groups and for the set of all mappings on some underlying finite set. But what about other, are there finite monoids that could not be written as the syntactic monoid of some language?


For a given automaton, by considering the monoid generated by the mappings induced by the letters on the states (the so called transformation monoid) when function composition is read from left to right, it holds that the transformation monoid of the minimal automaton is precisely the syntactic monoid. This observation helped me in constructing the above mentioned examples.

Let me also not that it is quite simple to realise any finite monoid $M$ as the transformation monoid of some automaton, simply take the elements of $M$ as the states, and consider every generator of $M$ as a letter of the alphabet and the transitions are given by $qx$ for some state $q$ and letter $x$, then the transformation monoid is isomorphic to $M$ itself (this is similar to the Cayley theorem about how groups embed into symmetric groups).
It seems there is a paper answering this exact question, and even in the more general case of $\omega$-regular languages, but I cannot find an open-access version. If somebody finds a link without paywall it would be great. I requested the full-text on ResearchGate.

Title: Which Finite Monoids are Syntactic Monoids of Rational omega-Languages.

Authors: Phan Trung Huy, Igor Litovsky, Do Long Van

Abstract: A notion of ω-rigid sets for a finite monoid is introduced. We prove that a finite monoid M is the Arnold's syntactic monoid of some rational ω-language (ω-syntactic for short) if and only if there exists an ω-rigid set for M. This property is shown to be decidable for the finite monoids. Relationship between the family of ω-syntactic monoids and that of ∗-syntactic monoids (i.e. the syntactic monoids of rational languages of finite words) is established.



Additionally, the wikipedia page on syntactic monoids states:


Every finite monoid is homomorphic to the syntactic monoid of some non-trivial language,[1] but not every finite monoid is isomorphic to a syntactic monoid.[2]
Every finite group is isomorphic to the syntactic monoid of some non-trivial language.[1]


[1] McNaughton, Robert; Papert, Seymour (1971). Counter-free Automata. Research Monograph 65. With an appendix by William Henneman. MIT Press. p. 48. ISBN 0-262-13076-9. Zbl 0232.94024.

[2] Lawson (2004) p.233