A Syntactical Proof of Locality of Da

Abstract

Using purely syntactical arguments, it is shown that every nontrivial pseudovariety of monoids contained in DO whose corresponding variety of languages is closed under unambiguous product, for instance DA, is local in the sense of Tilson.

The pseudoidentity problem and reducibility for completely regular semigroups

“…The adjacency functions are given by a(s) = 0(0(s)) and w(s) = l(l(s)) for edges s of content X and by a(s) = 0(s) and LJ(S) = l(s) for edges whose content misses a letter of A'. The product S\ o s 2 of consecutive edges s\, s 2 Similarly, we can show that u(sis 2 ) -w(s 2 ), while obviously a(sis' 2 ) = a(si). Hence the product o is well defined and it is easily verified that it is associative, thus showing that Sx is indeed a semigroupoid.…”

The globals of pseudovarieties of ordered semigroups containingB2and an application to a problem proposed by Pin

“…We note that in the proof of the Lemma 4.2(a), we show that if u is a path in Γ n and χ(u) may be factorized as χ(u) = u ′(1) 0 i 0 i+1 u ′(2) for some i, then there are appropriate paths u (1) and u (2) such that u = u (1) x i u (2) .…”

“…Arguing as in proof of Lemma 4.2(a) we obtain paths a and b in Γ n such that χ(a) = a ′′ and χ(b) = b ′′ . As it is observed above there are appropriate paths u (1) and u (2) such that u = u (1) x i−1 apbx j u (2) and v = u (1) x i−1 aqbx j u (2) . Since η 1 (p) = η 1 (q) it follows that η 1 (u) = η 1 (v).…”

“…Suppose now, without loss of generality, that v may be obtained from u applying a single rule p = q of L n , that is u = u (1) pu (2) and v = u (1) qu (2) , where u (i) (i = 1, 2) are appropriate paths in Γ n . Since χ is a homomorphism, Υ is a congruence, and (χ(p), χ(q)) ∈ Υ, it follows that (χ(u), χ(v)) ∈ Υ.…”

“…Without loss of generality, we may assume that (2) with b ′ = b ′′ 0 i ∈ A * i 0 i . So there are appropriate paths b, u (1) and u (2) such that u = u (1) x i−1 e i bx i u (2) and v = u (1) x i−1 bx i u (2) .…”