Regularity of monoids under Schutzenberger products

Homotopy bases and finite derivation type for Schutzenberger groups of monoids

Given a finitely presented monoid and a homotopy base for the monoid, and given an arbitrary Schutzenberger group of the monoid, the main result of this paper gives a homotopy base, and presentation, for the Schutzenberger group. In the case that the R-class R' of the Schutzenberger group G(H) has only finitely many H-classes, and there is an element s of the multiplicative right pointwise stabilizer of H, such that under the left action of the monoid on its R-classes the intersection of the orbit of the R-class of s with the inverse orbit of R' is finite, then finiteness of the presentation a…

On Residual Finiteness of Monoids, their Schutzenberger Groups and Associated Actions

In this paper we discuss connections between the following properties: (RFM) residual finiteness of a monoid M; (RFSG) residual finiteness of Schutzenberger groups of M; and (RFRL) residual finiteness of the natural actions of M on its Green's R- and L-classes. The general question is whether (RFM) implies (RFSG) and/or (RFRL), and vice versa. We consider these questions in all the possible combinations of the following situations: M is an arbitrary monoid; M is an arbitrary regular monoid; every J-class of M has finitely many R- and L-classes; M has finitely many left and right ideals. In eac…

On properties not inherited by monoids from their Schutzenberger groups

We give an example of a monoid with finitely many left and right ideals, all of whose Schutzenberger groups are presentable by finite complete rewriting systems, and so each have finite derivation type, but such that the monoid itself does not have finite derivation type, and therefore does not admit a presentation by a finite complete rewriting system. The example also serves as a counterexample to several other natural questions regarding complete rewriting systems and finite derivation type. Specifically it allows us to construct two finitely generated monoids M and N with isometric Cayley …

Decision problems for inverse monoids presented by a single sparse relator

We study a class of inverse monoids of the form M = Inv< X | w=1 >, where the single relator w has a combinatorial property that we call sparse. For a sparse word w, we prove that the word problem for M is decidable. We also show that the set of words in (X \cup X^{-1})^* that represent the identity in M is a deterministic context free language, and that the set of geodesics in the Schutzenberger graph of the identity of M is a regular language.

带对合的次塑性幺半群的表示与恒等式

Representations and identities of hypoplactic monoids with involution

本文研究带 Schützenberger 对合的次塑性幺半群，构造了其在上三角矩阵半环上的忠实表示，给出了字恒等式的组合刻画，并证明其非有限基当且仅当秩为2或3，同时给出判定字恒等式的多项式时间算法。

幺半群的扭曲积

Twisted products of monoids

本文研究幺半群扭曲积的结构，识别紧扭曲并刻画其 Green 关系、正则元、幂等元、极大子群等，并给出基于 Sylvester 秩不等式推广的新例子。