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1
Content available remote Polynomially representable semirings
EN
We characterize semirings which can be represented by an algebra of binary polynomials of the form a = x + y where the operations are compositions of functions. Furthermore, we classify, which algebras with two binary and two nullary operations (satisfying some natural identities) can be represented in this way, and how these algebras are related to semirings.
2
Content available remote Quantifiers on lattices with an antitone involution
EN
Quantifiers on lattices with an antitone involution are considered and it is proved that the poset of existential quantifiers is antiisomorphic to the poset of relatively complete sublattices.
3
Content available remote Characterizations of posets via weak states
EN
Weak states on posets are defined which are in some analogy to states on orthomodular posets used in axiomatic quantum mechanics. It is shown how certain properties of the set of weak states characterize certain properties of the underlying poset.
4
Content available remote When is a BCC-algebra equivalent to an MV-algebra?
EN
The aim of this paper is to characterize BCC-algebras which are term equivalent to MV-algebras. It turns out that they arę just the bounded commutative BCC-algebras. Purther, we characterize congruence kernels as deductive systems. The explicit description of a principal deductive system enables us to prove that every subdirectly irreducible bounded commutatwe BCC-algebra is a chain (with respect to the induced order) .
5
Content available remote Mv-like algebras associated to lambda-ortholattices
EN
The concept of a ambda-lattice generalizes a lattice by substituting associativity by the so-called skew associativity. When a bounded ambda-lattice is equipped with a monotonous unary involution which is a complementation, it is called a ambda-ortholattice. For ambda-ortholattices a Sheffer operation is constructed and, moreover, a derived algebra analogous to an MV-algebra is assigned whenever the ambda-lattice has antitone involutions on sections.
6
Content available remote A common generalization of ortholattices and Boolean quasirings
EN
In [2] a common generalization of Boolean algebras and Boolean rings was introduced. In a similar way we introduce a common generalization of ortholattices and Boolean quasirings.
7
Content available remote Weak fuzzy implication algebras and induced structures
EN
The concept of fuzzy implication algebra is weakened replacing the exchange axiom by an essentially weaker version. To each such algebra a groupoid is assigned. We get conditions under which this groupoid is commutative and show when a fuzzy implication algebra becomes a lattice with antitone involutions on all sections.
8
Content available remote SAI-lattices and ringoids
EN
The natural bijective correspondence between Boolean algebras and Boolean rings is generalized from Boolean algebras to lattices with 0 every principal ideal of which has an antitone involution. The corresponding ring-like structures are called ring-oids. Among them orthorings are characterized by a simple axiom. It is shown that congruences on ringoids are determined by their kernels and that ringoids are permutable at 0.
9
Content available remote Externalization of lattices
EN
Let r be a type of algebras. An identity s = t of type r is said to be externally compatible, or simply external, if the terms s and t are either the same variable or both start with the same operation symbol fj of the type. A variety is called external if all of its identities are external. For any variety V , there is a least external variety E(V ) containing V , the variety determined by the set of all external identities of V . External identities and varieties have been studied by [4], [5] and [2], and a general characterization of the algebras in E(V ) has been given in [3]. In this paper we study the algebras of the variety E(V ) where V is the type (2, 2) variety L of lattices. Algebras in L may also be described as ordered sets, and we give an ordered set description of the algebras in E(L). We show that on any algebra in E(L) there is a natural quasiorder having an additional property called externality, and that any set with such a quasiorder can be given the structure of an algebra in E(L). We also characterize algebras in E(L) by an inflation construction.
10
Content available remote Modifications of MV-algebras corresponding to strong ortholattices
EN
It is well-known that principal filters of MV-algebras are de Morgan algebras with involutory complementation. A modification of the notion of an MV-algebra is presented having the property that all principal filters are ortholattices. It turns out that the commutativity of these modified MV-algebras is equivalent to the distributivity of the corresponding ortholattices.
11
Content available remote A note on configurence uniformity for single algebras
EN
It is proved that though for varieties congruence uniformity implies congruence regularity this is not the case with infinite algebras.
12
Content available remote Ternary t-deductive systems
EN
The concept of deductive systems was introduced by A. Diego [6] in Hilbert algebras. For universal algebras, it was generalized in [3], and the ternary version was occured in [1] where a basic connection between ternary deductive systems and congruence c.lasses was established. An approach using Galois connections for binary deductive systems was developed in [4]. In the present paper, an approach similar to that of [4] is applied for a modified version of ternary deductive systems.
13
Content available remote Order algebras
EN
To every ordered set with a greatest element is assigned an order algebra, a Hilbert algebra occuring in intuicionistic logic. We prove some basic properties of order algebras and characterize their ideals and congruences. We introduce a concept of (relative) annihilator and show that it is a (relative) pseudocomplement in the lattice of all ideals.
14
Content available remote Regularity of generalized MV-algebras
EN
A generalized MV algebra is constructed by means of an 1-group m a way similar to that an MV-algebra is related to a commutative 1-group, see e.g. [11], [9]. We prove that the variety of generalized MV-algebras is congruence regular and give an explicit description of congruence classes.
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