Stephen J. Pride | Searcharxiv

arXiv2011

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 and of the homotopy base is preserved.

math.GR↗

arXiv2006

Homological finiteness conditions for groups, monoids and algebras

Recently Alonso and Hermiller introduced a homological finiteness condition\break $bi{-}FP_n$ (here called {\it weak} $bi{-}FP_n$) for monoid rings, and Kobayashi and Otto introduced a different property, also called $bi{-}FP_n$ (we adhere to their terminology). From these and other papers we know that: $bi{-}FP_n \Rightarrow$ left and right $FP_n \Rightarrow$ weak $bi{-}FP_n$; the first implication is not reversible in general; the second implication is reversible for group rings. We show that the second implication is reversible in general, even for arbitrary associative algebras (Theorem 1'), and we show that the first implication {\it is} reversible for group rings (Theorem 2). We also show that the all four properties are equivalent for connected graded algebras (Theorem 4). A result on retractions (Theorem 3') is proved, and some questions are raised.

math.GR↗