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EN
In Universal Algebra the structure of congruences for algebraic systems is fairly well investigated, and the relationship to the structure of the underlying system proper is well known. We propose a first step into this direction for studying the structure of congruences for stochastic relations. A Galois connection to a certain class of Boolean σ-algebras is exploited, atoms and antiatoms are identified, and it is show that a σ-basis exists. These constructions are applied to the problem of finding bisimulation cuts of a congruence. It cuts the relation through a span of morphisms with a minimum of joint events.
2
Sameness between based universal algebras
EN
This is the continuation of the paper "Transformations between Menger systems". To define when two universal algebras with bases "are the same", here we propose a universal notion of transformation that comes from a triple characterization concerning three representation facets: the determinations of the Menger system, analytic monoid and endomorphism representation corresponding to a basis. Hence, this notion consists of three equivalent definitions. It characterizes another technical variant and also the universal version of the very semi-linear transformations that were coordinate-free. Universal transformations allow us to check the actual invariance of general algebraic constructions, contrary to the seeming invariance of representation-free thinking. They propose a new interpretation of free algebras as superpositions of "analytic spaces" and deny that our algebras differ from vector spaces at fundamental stages. Contrary to present beliefs, even the foundation of abstract Linear Algebra turns out to be incomplete.
3
Subdirect product representations of some unary extensions of semilattices
EN
An algebra [...] represents the sequence so = (0, 3, l, l, . . .) if there are no constants in [...], there are exactly 3 distinct essentially unary polynomials in [...] and exactly l essentially n-ary polynomial in [...] for every n > l . It was proved in [4] that an algebra [..] represents the sequence so if and only if it is clone equivalent to a generic of one of three varieties V1, V2, V3, see Section l of [4]. Moreover, some representations of algebras from these varieties by means of semilattice ordered systems of algebras were given in [4] . In this paper we give another, by subdirect products, representation of algebras from V1, V2, V3. Moreover, we describe all subdirectly irreducible algebras from these varieties and we show that if an algebra [...] represents the sequence so, then it must be of cardinality at least 4.
4
Equational bases for k-normal identities
EN
The depth of a term may be used as a measurement of complexity of identities. For any natural number [...] have depth at least k. For any variety V, the k-normalization of V is the variety Nk(V) defined by all k-normal identities of V. We describe a process to produce from a basis for V a basis for Nk(V), for any variety V which has an idempotent term; when the type of V is finite and V is finitely based, this results in a finite basis for Nk(V) as well. This process encompasses several known examples, for varieties of bands and lattices, and allows us to give a new basis for the normalization of the variety PL of pseudo-complemented lattices.
5
Submonoids of generalized hypersubstitutions
EN
In this paper we define the operation G on the set of all generalized hypersubstitutions and investigate some algebraic-structural properties of the set of all generalized hypersubstitutions and of some submonoids M of the set of all generalized hypersubstitutions, respectively.
6
M-hyperquasivarieties
EN
We consider the notion of M-hyper-quasi-identities and M-hyperquasi-varieties, as a common generalization of the concept of quasi-identity (hyper-quasi-identity) and quasivariety (hyper-quasivariety) invented by A. I. Mal'cev, cf. [13], cf. [6] and hypervariety invented by the authors in [15], [8] and hy p erqu as i variety [9]. The results of this paper were presented on the 69th Workshop on General Algebra, held at Potsdam University (Germany) on March 18-20, 2005.
EN
In the paper the duality of the the notions of (higher) hypergraph and (higher) partition is shown. Both higher level hypergraphs and higher partitions are characterized algebr aically as left and right regular bands.
8
Yet another note on congruence uniformity
EN
We describe a countable unary algebra with few operations which is congruence uniform but not congruence regular, and we show that no uncountable algebra with these properties exists.
9
On the notion of homomorphism in many-sorted algebras
10
Commutative loops of exponent 3 with x.(x.y)2=y2
EN
It is well known that the class of Hall triple systems [5], Steiner triple systems in which each triangle generates an affine plane over GF(3), corresponds to the class of commutative Moufang loops of exponent 3 [6]. In this paper, we extend the class of algebras to the class of all commutative loops of exponent 3 satisfying the identity x.(x.y)2=y2, corresponding to the class of all Steiner triple systems. Such a commutative loop of exponent 3 with x . (x o y)2 = y2 is polynomially equivalent to a squag.
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