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The dimension of a variety V of algebras of a given type was introduced by E. Graczynska and D. Schweigert in [7] as the cardinality of the set of all derived varieties of V which are properly contained in V. In this paper, we characterize all solid varieties of dimensions 0, 1, and 2; prove that the dimension of a variety of finite type is at most N-0; give an example of a variety which has infinite dimension; and show that for every n is an element of N there is a variety with dimension n. Finally, we show that the dimension of a variety is related to the concept of the semantical kernel of a hypersubstitution and apply this connection to calculate the dimension of the class of all algebras of type tau = (n).
An n-ary cooperation is a mapping from a nonempty set A to the nth copower of A. A clone of cooperations is a set of cooperations which is closed under superposition and contains all injections. Coalgebras are pairs consisting of a set and a set of cooperations defined on this set. We define terms for coalgebras, coidentities and cohyperidentities. These concepts will be applied to give a new solution of the completeness problem for clones of cooperations defined on a two-element set and to separate clones of cooperations by coidentities.
Arenes with various alkyl side-chains were synthesized in high yields and excellent regioselectivities. Starting from toluic and naphthoic acids, the carboxylate group was conveniently substituted by alkyl halides by Birch reduction and subsequent decarbonylation. The method is characterized by inexpensive starting materials and reagents, and methylation of arenes was realized. Besides simple alkyl substituents, the scope of arene functionalization was extended by benzyl, fluoro, amino, and ester groups. We were able to control the alkylation of 1-naphthoic acid during Birch reduction by the addition of tert-butanol. This allowed the regioselective synthesis of mono and bis-substituted naphthalenes from the same starting material.
ie athenische Totenrede (epitaphios logos) spielte eine Schlüsselrolle bei der Konstruktion des idealisierten Selbstbildes des demokratischen Athen (5./4. Jh.v.Chr.). Dieser Aufsatz zeigt auf, dass die beiden bekanntesten Beispiele für das rhetorische Genre, die Rede des Perikes auf die Gefallenen bei Thukydides und die Rede des Sokrates im platonischen Dialog Menexenos, tatsächlich scharfe Kritik an der Instituton der Totenrede und an der demokratischen Polis an sich üben, obwohl beide Texte für sich betrachtet gemeinhin als affirmativ verstanden werden.