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Register a new userA new look at Ternary Boolean Algebras
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We present a new approach to ternary Boolean algebras in which negation is derived from the ternary operation. The key aspect is the replacement of complete commutativity by other axioms that do not require the ternary operation to be symmetric.
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In this article we investigate the notion and basic properties of Boolean algebras and prove the Stones representation theorem. The relations of Boolean algebras to logic and to set theory will be studied and, in particular, a neat proof of completeness theorem in propositional logic will be given using Stones theorem from Boolean algebra.
Boolean-type algebra (BTA) is investigated. A BTA is decomposed into Boolean-type lattice (BTL) and a complementation algebra (CA). When the object set is finite, the matrix expressions of BTL and CA (and then BTA) are presented. The construction and certain properties of BTAs are investigated via their matrix expression, including the homomorphism and isomorphism, etc. Then the product/decomposition of BTLs are considered. A necessary and sufficient condition for decomposition of BTA is obtained. Finally, a universal generator is provided for arbitrary finite universal algebras.
The article is a study of two algebraic structures, the `contrapositionally complemented pseudo-Boolean algebra (ccpBa) and `contrapositionally $vee$ complemented pseudo-Boolean algebra (c$vee$cpBa). The algebras have recently been obtained from a topos-theoretic study of categories of rough sets. The salient feature of these algebras is that there are two negations, one intuitionistic and another minimal in nature, along with a condition connecting the two operators. We study properties of these algebras, give examples, and compare them with relevant existing algebras. `Intuitionistic Logic with Minimal Negation (ILM) corresponding to ccpBas and its extension ILM-${vee}$ for c$vee$cpBas, are then investigated. Besides its relations with intuitionistic and minimal logics, ILM is observed to be related to Peirces logic. With a focus on properties of the two negations, two kinds of relational semantics for ILM and ILM-${vee}$ are obtained, and an inter-translation between the two semantics is provided. Extracting features of the two negations in the algebras, a further investigation is made, following logical studies of negations that define the operators independently of the binary operator of implication. Using Dunns logical framework for the purpose, two logics $K_{im}$ and $K_{im-{vee}}$ are presented, where the language does not include implication. $K_{im}$-algebras are reducts of ccpBas. The negations in the algebras are shown to occupy distinct positions in an enhanced form of Dunns Kite of negations. Relational semantics for $K_{im}$ and $K_{im-{vee}}$ are given, based on Dunns compatibility frames. Finally, relationships are established between the different algebraic and relational semantics for the logics defined in the work.
Making use of the modern techniques of non-holonomic geometry and constrained variational calculus, a revisitation of Ostrogradskys Hamiltonian formulation of the evolution equations determined by a Lagrangian of order >= 2 in the derivatives of the configuration variables is presented.
We use a new fiber spectroscopic survey of 12 nearby, poor groups of galaxies to examine the dynamics and evolution of galaxies in these common, but poorly studied, environments. Some of our conclusions are: (1) The nine groups in our sample with diffuse X-ray emission are in fact bound systems with at least 20-50 group members with absolute magnitudes as faint as M_B ~ -14 + 5 log h. (2) Galaxies in each X-ray-detected group have not all merged together because a significant fraction of the group mass lies outside of the galaxies and in a common halo, thereby reducing the rate of galaxy-galaxy interactions. (3) The similarity of the recent star formation histories and the fraction of early type galaxies in some poor groups to those in rich clusters suggests that cluster-specific environmental effects may not play a dominant role in the recent evolution of cluster galaxies. The evolution of group and cluster members may be driven instead by galaxy-galaxy interactions, which are likely to occur with equal frequency in field groups and in groups that have recently fallen into clusters (i.e., subclusters).
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