326 research outputs found
General coupled semirings of residuated lattices
Di Nola and Gerla showed that MV-algebras and coupled semirings are in a
natural one-to-one correspondence. We generalize this correspondence to
residuated lattices satisfying the double negation law
Coupled right orthosemirings induced by orthomodular lattices
L. P. Belluce, A. Di Nola and B. Gerla established a connection between
MV-algebras and (dually) lattice ordered semirings by means of so-called
coupled semirings. A similar connection was found for basic algebras and
semilattice ordered right near semirings by the authors. The aim of this paper
is to derive an analogous connection for orthomodular lattices and certain
semilattice ordered near semirings via so-called coupled right orthosemirings
Transition operators assigned to physical systems
By a physical system we recognize a set of propositions about a given system
with their truth-values depending on the states of the system. Since every
physical system can go from one state in another one, there exists a binary
relation on the set of states describing this transition. Our aim is to assign
to every such system an operator on the set of propositions which is fully
determined by the mentioned relation. We establish conditions under which the
given relation can be recovered by means of this transition operator
When does a semiring become a residuated lattice?
It is an easy observation that every residuated lattice is in fact a semiring
because multiplication distributes over join and the other axioms of a semiring
are satisfied trivially. This semiring is commutative, idempotent and simple.
The natural question arises if the converse assertion is also true. We show
that the conversion is possible provided the given semiring is, moreover,
completely distributive. We characterize semirings associated to complete
residuated lattices satisfying the double negation law where the assumption of
complete distributivity can be omitted. A similar result is obtained for
idempotent residuated lattices
Set Representation of Dynamic De Morgan algebras
By a De Morgan algebra is meant a bounded poset equipped with an antitone
involution considered as negation. Such an algebra can be considered as an
algebraic axiomatization of a propositional logic satisfying the double
negation law. Our aim is to introduce the so-called tense operators in every De
Morgan algebra for to get an algebraic counterpart of a tense logic with
negation satisfying the double negation law which need not be Boolean.
Following the standard construction of tense operators and by a frame
we solve the following question: if a dynamic De Morgan algebra is given, how
to find a frame such that its tense operators and can be reached by
this construction.Comment: 7 page
Residuated Relational Systems
The aim of the present paper is to generalize the concept of residuated
poset, by replacing the usual partial ordering by a generic binary relation,
giving rise to relational systems which are residuated. In particular, we
modify the definition of adjointness in such a way that the ordering relation
can be harmlessly replaced by a binary relation. By enriching such binary
relation with additional properties we get interesting properties of residuated
relational systems which are analogical to those of residuated posets and
lattices.Comment: 15 pages, pre-print version of a paper on Asian-European Journal of
Mathematic
Dynamic logic assigned to automata
A dynamic logic can be assigned to every automaton without regard if is deterministic or nondeterministic. This
logic enables us to formulate observations on in the form of
composed propositions and, due to a transition functor , it captures the
dynamic behaviour of . There are formulated conditions under
which the automaton can be recovered by means of
and .Comment: 12 pages. arXiv admin note: text overlap with arXiv:1510.0297
Involutive right-residuated l-groupoids
A common generalization of orthomodular lattices and residuated lattices is
provided corresponding to bounded lattices with an involution and sectionally
extensive mappings. It turns out that such a generalization can be based on
integral right-residuated l-groupoids. This general framework is applied to
MV-algebras,orthomodular lattices, Nelson algebras, basic algebras and Heyting
algebras.Comment: 22 page
Residuation in non-associative MV-algebras
It is well known that every MV-algebra can be converted into a residuated
lattice satisfying divisibility and the double negation law. In our previous
papers we introduced the concept of an NMV-algebra which is a non-associative
modification of an MV-algebra. The natural question arises if an NMV-algebra
can be converted into a residuated structure, too. Contrary to MV-algebras,
NMV-algebras are not based on lattices but only on directed posets and the
binary operation need not be associative and hence we cannot expect to obtain a
residuated lattice but only an essentially weaker structure called a
conditionally residuated poset. Considering several additional natural
conditions we show that every NMV-algebra can be converted in such a structure.
Also conversely, every such structure can be organized into an NMV-algebra.
Further, we study a bit more stronger version of an algebra where the binary
operation is even monotonous. We show that such an algebra can be organized
into a residuated poset and, conversely, every residuated poset can be
converted in this structure
On a variety of commutative multiplicatively idempotent semirings
We prove that the variety V of commutative multiplicatively idempotent
semirings satisfying x + y + xyz = x + y is generated by single semirings.
Moreover, we describe a normal form system for terms in V and we show that the
word problem in V is solvable. Although V is locally finite, it is residually
big
- …