Intervals in generalized effect algebras

A significant property of a generalized effect algebra is that its every interval with inherited partial sum is an effect algebra. We show that in some sense the converse is also true. More precisely, we prove that a set with zero element is a generalized effect algebra if and only if all its intervals are effect algebras. We investigate inheritance of some properties from intervals to generalized effect algebras, e.g., the Riesz decomposition property, compatibility of every pair of elements, dense embedding into a complete effect algebra, to be a sub-(generalized) effect algebra, to be lattice ordered and others. The response to the Open Problem from Rie?anová and Zajac (2013) for generalized effect algebras and their sub-generalized effect algebras is given.     

Isomorphism theorems on generalized effect algebras based on atoms

A well-known fact is that every generalized effect algebra can be uniquely extended to an effect algebra in which it becomes a sub-generalized effect algebra and simultaneously a proper order ideal, the set-theoretic complement of which is its dual poset. We show that two non-isomorphic generalized effect algebras (even finite ones) may have isomorphic effect algebraic extensions. For Archimedean atomic lattice effect algebras we prove “Isomorphism theorem based on atoms”. As an application we obtain necessary and sufficient conditions for isomorphism of two prelattice Archimedean atomic generalized effect algebras with common (or isomorphic) effect algebraic extensions.     

Torsion elements in effect algebras

We define the torsion element in effect algebras and use it to characterize MV-effect algebra and 0-homogeneous effect algebras in chain-complete effect algebras. As an application, we prove that every element of an orthocomplete homogeneous atomic effect algebra has a unique basic decomposition into a sum of a sharp element and unsharp multiples of atoms. Further, we characterize homogeneity by the set of all sharp elements in orthocomplete atomic effect algebras.     

Dynamic effect algebras and their representations

For lattice effect algebras, the so-called tense operators were already introduced by Chajda and Kola?ík. Tense operators express the quantifiers “it is always going to be the case that” and “it has always been the case that” and hence enable us to express the dimension of time in the logic of quantum mechanics. We present an axiomatization of these tense operators and prove that in every effect algebra can be introduced tense operators which, for non-complete lattice effect algebras, can be only partial mappings. An effect algebra equipped with tense operators reflects changes of quantum events from past to future. A crucial problem concerning tense operators is their representation. Having an effect algebra with tense operators, we can ask if there exists a frame such that each of these operators can be obtained by our construction. We solve this problem for (strict) dynamic effect algebras having a full set of homorphisms into a complete lattice effect algebra.     

Pseudocomplemented lattice effect algebras and existence of states

We prove that in every pseudocomplemented atomic lattice effect algebra the subset of all pseudocomplements is a Boolean algebra including the set of sharp elements as a subalgebra. As an application, we show families of effect algebras for which the existence of a pseudocomplementation implies the existence of states. These states can be obtained by smearing of states existing on the Boolean algebra of sharp elements.     

Synchronous Kleene algebra

The work presented here investigates the combination of Kleene algebra with the synchrony model of concurrency from Milner’s SCCS calculus. The resulting algebraic structure is called synchronous Kleene algebra. Models are given in terms of sets of synchronous strings and finite automata accepting synchronous strings. The extension of synchronous Kleene algebra with Boolean tests is presented together with models on sets of guarded synchronous strings and the associated automata on guarded synchronous strings. Completeness w.r.t. the standard interpretations is given for each of the two new formalisms. Decidability follows from completeness. Kleene algebra with synchrony should be included in the class of true concurrency models. In this direction, a comparison with Mazurkiewicz traces is made which yields their incomparability with synchronous Kleene algebras (one cannot simulate the other). On the other hand, we isolate a class of pomsets which captures exactly synchronous Kleene algebras. We present an application to Hoare-like reasoning about parallel programs in the style of synchrony.     

0-homogeneous effect algebras

In the present paper, we introduce a proper superclass of homogeneous effect algebras. We call this superclass as 0-homogeneous effect algebras. We prove that in every 0-homogeneous effect algebra, the set of all sharp elements forms a subalgebra. Every chain-complete 0-homogeneous effect algebra is homogeneous.     

Riesz ideals in generalized pseudo effect algebras and in their unitizations

We prove that there is an order isomorphism between the lattice of all normal Riesz ideals and the lattice of all Riesz congruences in upwards directed generalized pseudoeffect algebras (or GPEAs, for short). We give a sufficient and necessary condition under which a normal Riesz ideal I of a weak commutative generalized pseudoeffect algebra P is a normal Riesz ideal also in the unitization [^(P)]\widehat{P} of P. These results extend those obtained recently by Avalllone, Vitolo, Pulmannová and Vinceková for effect algebras. At the same time, we give the conditions under which the quotient of a generalized pseudoeffect algebra P is a generalized effect algebra and linearly ordered generalized pseudoeffect algebra.     

Growing certainty with null values

Non-availability of part of the data is a problem common to many database systems. We study here some aspects relating to incomplete information. Obviously, when the information in a database is not complete the answer to any query is only an approximation to the true result. The aim is to get a precise approximation. We regard databases as many-sorted algebras. Based on the concept of extended algebra we define what it means for an algebra to approximate another algebra. We then give the following simple principle for extending query languages to handle missing data: “Whenever information is added to an incomplete database subsequent answers to queries must not be contradictory or less informative than previously.” We then apply this principle to extend the functional query language Varqa. Finally, we compare the previously proposed many-valued logic systems with the system devised based on our principles.     

Sub-effect algebras and Boolean sub-effect algebras

 We show that Boolean effect algebras may have proper sub-effect algebras and conversely. Properties of lattice effect algebras with two blocks are shown. One condition of the completness of effect algebras is given. We also show that a lattice effect algebra associated to an orthomodular lattice can be embedded into a complete effect algebra iff the orthomodular lattice can be embedded into a complete orthomodular lattice.     

Rough 3-valued algebras

Rough set theory is an important tool for dealing with granularity and vagueness in information systems. This paper studies a kind of rough set algebra. The collection of all the rough sets of an approximation space can be made into a 3-valued Lukasiewicz algebra. We call the algebra a rough 3-valued Lukasiewicz algebra. In this paper, we focus on the rough 3-valued Lukasiewicz algebras, which are a special kind of 3-valued Lukasiewicz algebras. Firstly, we examine whether the rough 3-valued Lukasiewicz algebra is an axled 3-valued Lukasiewicz algebra. Secondly, we present the condition under which the rough 3-valued Lukasiewicz algebra is also a 3-valued Post algebra. Then we investigate the 3-valued Post subalgebra problem of the rough 3-valued Lukasiewicz algebra. Finally, this paper studies the relationship between the rough 3-valued Lukasiewicz algebra and the Boolean algebra constructed by all the exact sets of the corresponding approximation space.     

On BCK algebras: Part II: New algebras. The ordinal sum (product) of two bounded BCK algebras

Since all the algebras connected to logic have, more or less explicitly, an associated order relation, it follows, by duality principle, that they have two presentations, dual to each other. We classify these dual presentations in “left” and “right” ones and we consider that, when dealing with several algebras in the same research, it is useful to present them unitarily, either as “left” algebras or as “right” algebras. In some circumstances, this choice is essential, for instance if we want to build the ordinal sum (product) between a BL algebra and an MV algebra. We have chosen the “left” presentation and several algebras of logic have been redefined as particular cases of BCK algebras. We introduce several new properties of algebras of logic, besides those usually existing in the literature, which generate a more refined classification, depending on the properties satisfied. In this work (Parts I–V) we make an exhaustive study of these algebras—with two bounds and with one bound—and we present classes of finite examples, in bounded case. In Part II, we continue to present new properties, and consequently new algebras; among them, bounded α γ algebra is a common generalization of MTL algebra and divisible bounded residuated lattice (bounded commutative Rl-monoid). We introduce and study the ordinal sum (product) of two bounded BCK algebras. Dedicated to Grigore C. Moisil (1906–1973).     

E-perfect effect algebras

The definition of E-perfect effect algebras is introduced, and their structure is studied. We study the lexicographical product of an effect algebra with any upwards directed partially ordered Abelian group, and we show that every E-perfect effect algebra is isomorphic with such a kind of the lexicographical product.     

Embeddings of generalized effect algebras into complete effect algebras

Generalized effect algebras as posets are unbounded versions of effect algebras having bounded effect-algebraic extensions. We show that when the MacNeille completion MC(P) of a generalized effect algebra P cannot be organized into a complete effect algebra by extending the operation ⊕ onto MC(P) then still P may be densely embedded into a complete effect algebra. Namely, we show these facts for Archimedean GMV-effect algebras and block-finite prelattice generalized effect algebras. Moreover, we show that extendable commutative BCK-algebras directed upwards are equivalent to generalized MV-effect algebras.     

Identities for the Associator in Alternative Algebras

The associator is an alternating trilinear product for any alternative algebra. We study this trilinear product in three related algebras: the associator in a free alternative algebra, the associator in the Cayley algebra, and the ternary cross product on four-dimensional space. This last example is isomorphic to the ternary subalgebra of the Cayley algebra which is spanned by the non-quaternion basis elements. We determine the identities of degree  ≤  7 satisfied by these three ternary algebras. We discover two new identities in degree 7 satisfied by the associator in every alternative algebra and five new identities in degree 7 satisfied by the associator in the Cayley algebra. For the ternary cross product we recover the ternary derivation identity in degree 5 introduced by Filippov.     

BL-algebras and effect algebras

Although the notions of a BL-algebra and of an effect algebra arose in rather different contextes, both types of algebras have certain structural properties in common. To clarify their mutual relation, we introduce weak effect algebras, which generalize effect algebras in that the order is no longer necessarily determined by the partial addition. A subclass of the weak effect algebras is shown to be identifiable with the BL-algebras. Moreover, weak D-posets are defined, being based on a partial difference rather than a partial addition. They are equivalent to weak effect algebras. Finally, it is seen to which subclasses of the weak effect algebras certain subclasses of the BL-algebras, namely the MV-, product, and Gödel algebras, correspond.     

The equational theory of Kleene lattices

Languages and families of binary relations are standard interpretations of Kleene algebras. It is known that the equational theories of these interpretations coincide and that the free Kleene algebra is representable both as a relation and as a language algebra. We investigate the identities valid in these interpretations when we expand the signature of Kleene algebras with the meet operation. In both cases, meet is interpreted as intersection. We prove that in this case, there are more identities valid in language algebras than in relation algebras (exactly three more in some sense), and representability of the free algebra holds for the relational interpretation but fails for the language interpretation. However, if we exclude the identity constant from the algebras when we add meet, then the equational theories of the relational and language interpretations remain the same, and the free algebra is representable as a language algebra, too. The moral is that only the identity constant behaves differently in the language and the relational interpretations, and only meet makes this visible.     

Lattice pseudoeffect algebras as double residuated structures

Pseudoeffect algebras are partial algebraic structures which are non-commutative generalizations of effect algebras. The main result of the paper is a characterization of lattice pseudoeffect algebras in terms of so-called pseudo Sasaki algebras. In contrast to pseudoeffect algebras, pseudo Sasaki algebras are total algebras. They are obtained as a generalization of Sasaki algebras, which in turn characterize lattice effect algebras. Moreover, it is shown that lattice pseudoeffect algebras are a special case of double CI-posets, which are algebraic structures with two pairs of residuated operations, and which can be considered as generalizations of residuated posets. For instance, a lattice ordered pseudoeffect algebra, regarded as a double CI-poset, becomes a residuated poset if and only if it is a pseudo MV-algebra. It is also shown that an arbitrary pseudoeffect algebra can be described as a special case of conditional double CI-poset, in which case the two pairs of residuated operations are only partially defined.     

Regular Gröbner Bases

In this paper we introduce the concept of bi-automaton algebras, generalizing the automaton algebras previously defined by Ufnarovski. A bi-automaton algebra is a quotient of the free algebra, defined by a binomial ideal admitting a Gröbner basis which can be encoded as a regular set; we call such a Gröbner basis regular. We give several examples of bi-automaton algebras, and show how automata connected to regular Gröbner bases can be used to perform reduction.     

序列效应代数的Holland理论

介绍了序列效应代数的概念,给出了交换的序列效应代数的一些性质,证明了交换的序列效应代数由素理想诱导的商代数仍然是交换的序列效应代数。最后证明了每一个交换的序列效应代数能被表示为某个反格的自态射形成的序列效应代数。这样的表示是有用的,因为它给出了一个交换的序列效应代数作为自态射的集合的具体化。