Simulations of weighted tree automata (wta) are considered. It is shown how such simulations can be decomposed into simpler functional and dual functional simulations also called forward and backward simulations. In addition, it is shown in several cases (fields, commutative rings, Noetherian semirings, semiring of natural numbers) that all equivalent wta M and N can be joined by a finite chain...

The concept of (A,B)-invariant subspace is the fundamental concept of the geometric approach of control design. It has been extended by many authors to that of (A,B)-invariant module or semimodule, for the sake of extending the solution of various control problems to the case of systems over rings or semi rings. In this paper is discussed the use of dynamic feedback control laws for systems ove...

where the summation is over all injections σ from M . = {1, 2, . . . ,m} to N . = {1, 2, . . . , n}. While studies on permanents – since their introduction in 1812 by Binet [3] and Cauchy [5] – have focused on matrices over fields and commutative rings, we generally only assume the entries are from some semiring, that is, multiplication need not commute and additive inverses need not exist. In ...

Some public-key cryptosystems based on the tropical semiring have been proposed in recent years because of their increased efficiency, since multiplication is actually an ordinary addition numbers and there no semiring. However, most these security defects they adopt a public matrix to construct commutative semirings. This paper proposes new circular matrices. The relies NP-hard problem solving...

The Matrix-Tree Theorem is a well-known combinatorial result relating the value of the minors of a certain square matrix to the sum of the weights of the arborescences (= rooted directed trees) in the associated graph. We prove an extension of this result to algebraic structures much more general than the field of real numbers, namely commutative semirings. In such structures, the first law (ad...

A reconstruction problem is formulated for multisets over commutative groupoids. The cards of a multiset are obtained by replacing a pair of its elements by their sum. Necessary and sufficient conditions for the reconstructibility of multisets are determined. These results find an application in a different kind of reconstruction problem for functions of several arguments and identification min...

An open question in [FÜLÖP, MALETTI, VOGLER: Weighted extended tree transducers. Fundamenta Informaticae 111(2), 2011] asks whether weighted linear extended tree transducers preserve recognizability in countably complete commutative semirings. In this contribution, the question is answered positively, which is achieved with a construction that utilizes inside weights. Due to the completeness of...

An atomic monoid $M$ is called length-factorial if for every non-invertible element $x \in M$, no two distinct factorizations of $x$ into irreducibles have the same length (i.e., number irreducible factors, counting repetitions). The notion length-factoriality was introduced by J. Coykendall and W. Smith in 2011 under term 'other-half-factoriality': they used to provide a characterization uniqu...

An algebra is entropic if its basic operations are homomorphisms. The paper is focused on representations of such algebras. We prove the following theorem: An entropic algebra without constant basic operations which satisfies so called Szendrei identities and such that all its basic operations of arity at least two are surjective is a subreduct of a semimodule over a commutative semiring. Our t...