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331
On Defining Functions on Trees
, 2003
"... this paper. 1. PRELIMINARIES One can prove the following propositions: (1) For every non empty set D holds every finite sequence of elements of FinTrees(D) is a finite sequence of elements of Trees(D) ..."
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Cited by 50 (26 self)
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this paper. 1. PRELIMINARIES One can prove the following propositions: (1) For every non empty set D holds every finite sequence of elements of FinTrees(D) is a finite sequence of elements of Trees(D)
Subalgebras of many sorted algebra. Lattice of subalgebras
- Journal of Formalized Mathematics
, 1994
"... notation and terminology for this paper. ..."
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Transpose Matrices and Groups of Permutations
, 2003
"... Some facts concerning matrices with dimension 2 × 2 are shown. Upper and lower triangular matrices, and operation of deleting rows and columns in a matrix are introduced. Besides, we deal with sets of permutations and the fact that all permutations of finite set constitute a finite group is proved. ..."
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Cited by 34 (0 self)
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Some facts concerning matrices with dimension 2 × 2 are shown. Upper and lower triangular matrices, and operation of deleting rows and columns in a matrix are introduced. Besides, we deal with sets of permutations and the fact that all permutations of finite set constitute a finite group is proved. Some proofs are based on [11] and [14].
Sets and functions of trees and joining operations of trees
- Journal of Formalized Mathematics
, 1992
"... Summary. In the article we deal with sets of trees and functions yielding trees. So, we introduce the sets of all trees, all finite trees and of all trees decorated by elements from some set. Next, the functions and the finite sequences yielding (finite, decorated) trees are introduced. There are sh ..."
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Cited by 30 (15 self)
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Summary. In the article we deal with sets of trees and functions yielding trees. So, we introduce the sets of all trees, all finite trees and of all trees decorated by elements from some set. Next, the functions and the finite sequences yielding (finite, decorated) trees are introduced. There are shown some convenient but technical lemmas and clusters concerning with those concepts. In the fourth section we deal with trees decorated by Cartesian product and we introduce the concept of a tree called a substitution of structure of some finite tree. Finally, we introduce the operations of joining trees, i.e. for the finite sequence of trees we define the tree which is made by joining the trees from the sequence by common root. For one and two trees there are introduced the same operations.
The product and the determinant of matrices with entries in a field
, 2003
"... Concerned with a generalization of concepts introduced in [14], i.e. there are introduced the sum and the product of matrices of any dimension of elements of any field. ..."
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Cited by 27 (0 self)
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Concerned with a generalization of concepts introduced in [14], i.e. there are introduced the sum and the product of matrices of any dimension of elements of any field.
Semigroup operations on finite subsets
- Journal of Formalized Mathematics
, 1990
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Associated Matrix of Linear Map
, 2002
"... this paper. 1. PRELIMINARIES Let A be a set, let X be a set, let D be a non empty set of finite sequences of A, let p be a partial function from X to D, and let i be a set. Then p i is an element of D ..."
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Cited by 22 (1 self)
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this paper. 1. PRELIMINARIES Let A be a set, let X be a set, let D be a non empty set of finite sequences of A, let p be a partial function from X to D, and let i be a set. Then p i is an element of D
