A sufficient and necessary condition about regularity of a language

Here are some necessary and sufficient conditions for a language to be regular. Theorem. Let $LsubseteqSigma^*$. The following conditions are equivalent:

• $L$ is generated by a regular expression (i.e., the definition of regular language).
• $L$ is recognized by a nondeterministic finite automaton (Kleene).
• $L$ is recognized by a nondeterministic finite automaton without $varepsilon$-transitions.
• $L$ is recognized by a deterministic finite automaton (Scott and Rabin).
• $L$ is generated by a grammar $(N,Sigma,P,S)$, where $N$ is a finite subset of $Sigma^*$ (Frazier and Page).
• $L$ is generated by a left (resp. right) regular context-free grammar.
• The index of the Nerode relation $equiv_L$ is finite (Anil Nerode, Linear automaton transformations, 1958). This is widely (and incorrectly) known as the Myhill-Nerode theorem. $equiv_L$ is the relation used to build the minimal DFA of a regular language.
• The index of the Myhill relation $sim_L$ is finite (John Myhill, Finite Automata and the Representation of Events, 1957). $sim_L$ is the relation used to build the syntactic monoid of an arbitrary language.
• The syntactic monoid of $L$ is finite (consequence of Myhill’s result). We note here that the syntactic monoid, apart from being defined by using relation $sim_L$, can be defined as a minimal monoid (in size) that recognizes $L$ as a preimage of a homomorphism.
• $L$ can be recognized by a read-only Turing Machine (trivial).
• $L$ can be defined by a formula in Monadic second-order logic over strings (Büchi).

If a language does not satisfy the conditions of the pumping lemma for regular languages, then it is not regular. This means pumping lemma is a sufficient condition for the non-regularity of a language. In summary, statements 1, 2 and 3 are false, while statement 4 is true, as you mentioned.