Relative locations of subwords in free operated semigroups and Motzkin words

Abstract

Abstract. Bracketed words are basic structures both in mathematics (such as Rota-Baxter algebras) and mathematical physics (such as rooted trees) where the locations of the substructures are important. In this paper we give the classification of the relative locations of two bracketed subwords of a bracketed word in an operated semigroup into the separated, nested and intersecting cases. We achieve this by establishing a correspondence between relative locations of bracketed words and those of words by applying the concept of Motzkin words which are the algebraic forms of Motzkin paths.

Free integro-differential algebras and Gröbner–Shirshov bases

“…we have kR(∆X) kM {P} (∆X)/J RB kM {P,d} (X)/J DRB . By [24,Theorem 4.11], the statement of the present theorem holds when R n is replaced by M {P} (∆X). Since R(∆X) and hence R n are subsets of M {P} (∆X), the statement of the theorem remains true for R(∆X) and R n .…”

“…For the purpose of the proof the next lemma, we describe the relative location of two bracketed subwords in the more precise notion of placements (or occurrences [10]) in a bracketed word. See [24] for details. But note that we focus on words in R n as a subset of M(∆ n X).…”

“…So we have (24) ( f, g) w = f − q| g = − P(u)p| P(d(r)P(s)) + P(up| P(d(r)P(s)) ) + λP(ud(p| P(d(r)P(s)) )) + P(P(u)d(p| rP(s) )) − P(P(u)d(p| P(rs) )) − λP(P(u)d(p| P (d(r)s) )).…”

Free operated monoids and rewriting systems

“…The following concept is finer than bracketed subwords, containing the information of placements [29]. Definition 3.4.…”

Rota-Baxter type operators, rewriting systems and Gröbner-Shirshov bases

“…From the definition of n for n 0, we see that the restriction of the order n+1 to M n (Z) equals to the order n . Thus we can define the order (43) db := lim…”