The relationship between classes of tree-to-tree-series and o-tree-to-tree-series transformations, which are computed by restricted deterministic bottom-up weighted tree transducers, is investigated. Essentially, these transducers are deterministic bottom-up tree series transducers, except that the former are defined over monoids whereas the latter are defined over semirings and only use the mu...

For which groups G is it true that whenever one forms a direct limit of left G-sets, lim −→ i∈I Xi, the set of its fixed points, (lim −→I Xi) , can be obtained as the direct limit lim −→I (X i ) of the fixed point sets of the given G-sets? An easy argument shows that this holds if and only if G is finitely generated. If we replace “group G ” by “monoid M”, the answer is the less familiar condit...

For which groups G is it true that whenever one forms a direct limit of left G-sets, lim −→ i∈I Xi, the set of its fixed points, (lim −→I Xi) , can be obtained as the direct limit lim −→I (X i ) of the fixed point sets of the given G-sets? An easy argument shows that this is the case if and only if G is finitely generated. If we replace “group G” by “monoid M”, the answer is the less familiar c...

For a finite lattice L, the congruence lattice ConL of L can be easily computed from the partially ordered set J(L) of join-irreducible elements of L and the join-dependency relation DL on J(L). We establish a similar version of this result for the dimension monoid DimL of L, a natural precursor of ConL. For L join-semidistributive, this result takes the following form: Theorem 1. Let L be a fi...

We present three results on divisibility monoids. These divisibility monoids were introduced in [11] as an algebraic generalization of Mazurkiewicz trace monoids. (1) We give a decidable class of presentations that gives rise precisely to all divisibility monoids. (2) We show that any divisibility monoid is an automatic monoid [5]. This implies that its word problem is solvable in quadratic tim...

The main result of this paper gives a presentation for an arbitrary subgroup of a monoid defined by a presentation. It is a modification of the well known Reidemeister–Schreier theorem for groups. Some consequences of this result are explored. It is proved that a regular monoid with finitely many left and right ideals is finitely presented if and only if all its maximal subgroups are finitely p...

A monoid M is a Cale monoid with base Q if for every nonunit x ∈ M there exists a positive integer n such that xn factors uniquely up to order and associates as elements from Q ⊆ M\M. An integral domain D is a Cale domain with base Q, if its multiplicative monoid of nonzero elements is a Cale monoid with base Q. We explore the basic properties of Cale monoids and integral domains. In particular...

Following Buchberger's approach to computing a Grr obner basis of a polynomial ideal in polynomial rings, a completion procedure for nitely generated right ideals in ZH] is given, where H is an ordered monoid presented by a nite, convergent semi{Thue system ((; T). Taking a nite set F ZH] we get a (possibly innnite) basis of the right ideal generated by F , such that using this basis we have un...

The firing rule of Petri nets relies on a residuation operation for the commutative monoid of natural numbers. We identify a class of residuated commutative monoids, called Petri algebras, for which one can mimic the token game of Petri nets to define the behaviour of generalized Petri net whose flow relation and place contents are valued in such algebraic structures. We show that Petri algebra...

This paper proves that any monoid presented by a confluent context-free monadic rewriting system is word-hyperbolic. This result then applied to answer a question asked by Duncan&Gilman by exhibiting an example of a word-hyperbolic monoid that does not admit a word-hyperbolic structure with uniqueness (that is, in which the language of representatives maps bijectively onto the monoid).