http://hdl.handle.net/11320/21336 Thu, 29 Oct 2026 23:57:20 GMT 2026-10-29T23:57:20Z http://hdl.handle.net/11320/21342 Tytuł: Basic Properties of Determinants of Square Matrices over a Field Autorzy: Pąk, Karol Abstrakt: In this paper I present basic properties of the determinant of square matrices over a field and selected properties of the sign of a permutation. First, I define the sign of a permutation by the requirement where p is any fixed permutation of a set with n elements. I prove that the sign of a product of two permutations is the same as the product of their signs and show the relation between signs and parity of permutations. Then I consider the determinant of a linear combination of lines, the determinant of a matrix with permutated lines and the determinant of a matrix with a repeated line. Finally, at the end I prove that the determinant of a product of two square matrices is equal to the product of their determinants. Mon, 01 Jan 2007 00:00:00 GMT http://hdl.handle.net/11320/21342 2007-01-01T00:00:00Z http://hdl.handle.net/11320/21341 Tytuł: Formal Languages - Concatenation and Closure Autorzy: Trybulec, Michał Abstrakt: Formal languages are introduced as subsets of the set of all 0-based finite sequences over a given set (the alphabet). Concatenation, the n -th power and closure are defined and their properties are shown. Finally, it is shown that the closure of the alphabet (understood here as the language of words of length 1) equals to the set of all words over that alphabet, and that the alphabet is the minimal set with this property. Notation and terminology were taken from [5] and [13]. Mon, 01 Jan 2007 00:00:00 GMT http://hdl.handle.net/11320/21341 2007-01-01T00:00:00Z http://hdl.handle.net/11320/21337 Tytuł: Several Classes of BCI-algebras and their Properties Autorzy: Ding, Yuzhong Abstrakt: I have formalized the BCI-algebras closely following the book [6], sections 1.1 to 1.3, 1.6, 2.1 to 2.3, and 2.7. In this article the general theory of BCI-algebras and several classes of BCI-algebras are given. Mon, 01 Jan 2007 00:00:00 GMT http://hdl.handle.net/11320/21337 2007-01-01T00:00:00Z