586 research outputs found

### Fully representable and *-semisimple topological partial *-algebras

We continue our study of topological partial *-algebras, focusing our
attention to *-semisimple partial *-algebras, that is, those that possess a
{multiplication core} and sufficiently many *-representations. We discuss the
respective roles of invariant positive sesquilinear (ips) forms and
representable continuous linear functionals and focus on the case where the two
notions are completely interchangeable (fully representable partial *-algebras)
with the scope of characterizing a *-semisimple partial *-algebra. Finally we
describe various notions of bounded elements in such a partial *-algebra, in
particular, those defined in terms of a positive cone (order bounded elements).
The outcome is that, for an appropriate order relation, one recovers the
\M-bounded elements introduced in previous works.Comment: 26 pages, Studia Mathematica (2012) to appea

### Disjoint-union partial algebras

Disjoint union is a partial binary operation returning the union of two sets
if they are disjoint and undefined otherwise. A disjoint-union partial algebra
of sets is a collection of sets closed under disjoint unions, whenever they are
defined. We provide a recursive first-order axiomatisation of the class of
partial algebras isomorphic to a disjoint-union partial algebra of sets but
prove that no finite axiomatisation exists. We do the same for other signatures
including one or both of disjoint union and subset complement, another partial
binary operation we define.
Domain-disjoint union is a partial binary operation on partial functions,
returning the union if the arguments have disjoint domains and undefined
otherwise. For each signature including one or both of domain-disjoint union
and subset complement and optionally including composition, we consider the
class of partial algebras isomorphic to a collection of partial functions
closed under the operations. Again the classes prove to be axiomatisable, but
not finitely axiomatisable, in first-order logic.
We define the notion of pairwise combinability. For each of the previously
considered signatures, we examine the class isomorphic to a partial algebra of
sets/partial functions under an isomorphism mapping arbitrary suprema of
pairwise combinable sets to the corresponding disjoint unions. We prove that
for each case the class is not closed under elementary equivalence.
However, when intersection is added to any of the signatures considered, the
isomorphism class of the partial algebras of sets is finitely axiomatisable and
in each case we give such an axiomatisation.Comment: 30 page

### Noncommutativity as a colimit

Every partial algebra is the colimit of its total subalgebras. We prove this
result for partial Boolean algebras (including orthomodular lattices) and the
new notion of partial C*-algebras (including noncommutative C*-algebras), and
variations such as partial complete Boolean algebras and partial AW*-algebras.
The first two results are related by taking projections. As corollaries we find
extensions of Stone duality and Gelfand duality. Finally, we investigate the
extent to which the Bohrification construction, that works on partial
C*-algebras, is functorial.Comment: 22 pages; updated theorem 15, added propoisition 3

### NeutroAlgebra is a Generalization of Partial Algebra

In this paper we recall, improve, and extend several definitions, properties and applications of our previous 2019
research referred to NeutroAlgebras and AntiAlgebras (also called NeutroAlgebraic Structures and respectively
AntiAlgebraic Structures).
Let \u3cA\u3e be an item (concept, attribute, idea, proposition, theory, etc.). Through the process of neutrosphication, we
split the nonempty space we work on into three regions {two opposite ones corresponding to \u3cA\u3e and \u3cantiA\u3e, and
one corresponding to neutral (indeterminate) \u3cneutA\u3e (also denoted \u3cneutroA\u3e) between the opposites}, which may
or may not be disjoint â€“ depending on the application, but they are exhaustive (their union equals the whole space).
A NeutroAlgebra is an algebra which has at least one NeutroOperation or one NeutroAxiom (axiom that is true for
some elements, indeterminate for other elements, and false for the other elements).
A Partial Algebra is an algebra that has at least one Partial Operation, and all its Axioms are classical (i.e. axioms true
for all elements).
Through a theorem we prove that NeutroAlgebra is a generalization of Partial Algebra, and we give examples of
NeutroAlgebras that are not Partial Algebras. We also introduce the NeutroFunction (and NeutroOperation)

### Rewriting in the partial algebra of typed terms modulo AC

AbstractWe study the partial algebra of typed terms with an associative commutative and idempotent operator (typed AC-terms). The originality lies in the representation of the typing policy by a graph which has a decidable monadic theory.In this paper we show on two examples that some results on AC-terms can be raised to the level of typed AC-terms. The examples are the results on rational languages (in particular their closure by complement) and the property reachability problem for ground rewrite systems (equivalently process rewrite systems)

### Congruence Lattices of Certain Finite Algebras with Three Commutative Binary Operations

A partial algebra construction of Gr\"atzer and Schmidt from
"Characterizations of congruence lattices of abstract algebras" (Acta Sci.
Math. (Szeged) 24 (1963), 34-59) is adapted to provide an alternative proof to
a well-known fact that every finite distributive lattice is representable, seen
as a special case of the Finite Lattice Representation Problem.
The construction of this proof brings together Birkhoff's representation
theorem for finite distributive lattices, an emphasis on boolean lattices when
representing finite lattices, and a perspective based on inequalities of
partially ordered sets. It may be possible to generalize the techniques used in
this approach.
Other than the aforementioned representation theorem only elementary tools
are used for the two theorems of this note. In particular there is no reliance
on group theoretical concepts or techniques (see P\'eter P\'al P\'alfy and
Pavel Pud\'lak), or on well-known methods, used to show certain finite lattice
to be representable (see William J. DeMeo), such as the closure method

### Representable linear functionals on partial *-algebras

A GNS - like *-representation of a \pa\ \A defined by certain representable
linear functionals on \A is constructed. The study of the interplay with the
GNS construction associated with invariant positive sesquilinear forms (ips)
leads to the notions of pre-core and of singular form. It is shown that a
positive sesquilinear form with pre-core always decomposes into the sum of an
ips form and a singular one

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