In this contribution the Myhill-Nerode congruence relation on tree series is reviewed and a more detailed analysis of its properties is presented. It is shown that, if a tree series is deterministically recognizable over a zero-divisor free and commutative semiring, then the Myhill-Nerode congruence relation has finite index. By [Borchardt: Myhill-Nerode Theorem for Recognizable Tree Series. LN...

Let M be a commutative cancellative atomic monoid. We consider the behavior of the asymptotic length functions ̄̀(x) and L̄(x) on M . If M is finitely generated and reduced, then we present an algorithm for the computation of both ̄̀(x) and L̄(x) where x is a nonidentity element of M . We also explore the values that the functions ̄̀(x) and L̄(x) can attain when M is a Krull monoid with torsion diviso...

The concept of strong spined product of semigroups is introduced. We first show that a semigroup S is a rpp-semigroup with left central idempotents if and only if S is a strong semilattice of left cancellative right stripes. Then, we show that such kind of semigroups can be described by the strong spined product of a C-rpp-semigroup and a right normal band. In particular, we show that a semigro...

We consider the class of “well-tempered” integer-valued scoring games, which have the property that the parity of the length of the game is independent of the line of play. We consider disjunctive sums of these games, and develop a theory for them analogous to the standard theory of disjunctive sums of normal-play partizan games. We show that the monoid of well-tempered scoring games modulo ind...

We introduce the growth partition function ZΓ,G(t) associated with any cancellative infinite monoid Γ with a finite generator system G. It is a power series in t whose coefficients lie in integral Lie-like space LZ(Γ, G) in the configuration algebra associated with the Cayley graph (Γ, G). We determine them for homogeneous monoids admitting left greatest common divisor and right common multiple...

Let $Q$ be an inverse semigroup. A subsemigroup $S$ of $Q$ is a left I-order in $Q$ and $Q$ is a semigroup of left I-quotients of $S$ if every element $qin Q$ can be written as $q=a^{-1}b$ for some $a,bin S$. If we insist on $a$ and $b$ being $er$-related in $Q$, then we say that $S$ is straight in $Q$. We characterize semigroups which are left I-quotients of left regular bands of right cancell...