http://hdl.handle.net/11320/21057 2026-09-18T12:31:13Z http://hdl.handle.net/11320/21070 Tytuł: Basic Properties and Concept of Selected Subsequence of Zero Based Finite Sequences Autorzy: Nakamura, Yatsuka; Ito, Hisashi Abstrakt: Here, we develop the theory of zero based finite sequences, which are sometimes, more useful in applications than normal one based finite sequences. The fundamental function Sgm is introduced as well as in case of normal finite sequences and other notions are also introduced. However, many theorems are a modification of old theorems of normal finite sequences, they are basically important and are necessary for applications. A new concept of selected subsequence is introduced. This concept came from the individual Ergodictheorem (see [7]) and it is the preparation for its proof. 2008-01-01T00:00:00Z http://hdl.handle.net/11320/21069 Tytuł: Orthomodular Lattices Autorzy: Mądra, Elżbieta; Grabowski, Adam Abstrakt: The main result of the article is the solution to the problem of short axiomatizations of orthomodular ortholattices. Based on EQP/Otter results [10], we gave a set of three equations which is equivalent to the classical, much longer equational basis of such a class. Also the basic example of the lattice which is not orthomodular, i.e. benzene (or B6) is defined in two settings– as a relational structure (poset) and as a lattice. As a preliminary work, we present the proofs of the dependence of other axiomatizations of ortholattices. The formalization of the properties of orthomodular lattices follows [4]. 2008-01-01T00:00:00Z http://hdl.handle.net/11320/21068 Tytuł: Linear Map of Matrices Autorzy: Pąk, Karol Abstrakt: The paper is concerned with a generalization of concepts introduced in [13], i.e. introduced are matrices of linear transformations over a finite dimensional vector space. Introduced are linear transformations over a finited imensional vector space depending on a given matrix of the transformation. Finally, I prove that the rank of linear transformations over a finite-dimensional vector space is the same as the rank of the matrix of that transformation. 2008-01-01T00:00:00Z http://hdl.handle.net/11320/21067 Tytuł: Block Diagonal Matrices Autorzy: Pąk, Karol Abstrakt: In this paper I present basic properties of block diagonal matrices over a set. In my approach the finite sequence of matrices in a block diagonal matrix is not restricted to square matrices. Moreover, the off-diagonal blocks need not be zero matrices, but also with another arbitrary fixed value. 2008-01-01T00:00:00Z