↑ 1. ⇒ 2. by Thompson's construction algorithm ↑ 2. ⇒ 1. by Kleene's algorithm or using Arden's lemma ↑ 2. ⇒ 3. by the powerset construction ↑ 3. ⇒ 2. since the former definition is stronger than the latter ↑ 2. ⇒ 4. see Hopcroft, Ullman (1979), Theorem 9.2, p.219 ↑ 4. ⇒ 2. see Hopcroft, Ullman (1979), Theorem 9.1, p.218 ↑ 3. ⇔ 10. by the Myhill–Nerode theorem ↑ u~v is defined as: uw∈L if and only if vw∈L for all w∈Σ* ↑ 3. ⇔ 11. see the proof in the Syntactic monoid article, and see p.160 in Holcombe, W.M.L. (1982). Algebraic automata theory. Cambridge Studies in Advanced Mathematics.