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30
Many sorted algebras
 Journal of Formalized Mathematics
, 1994
"... Summary. The basic purpose of the paper is to prepare preliminaries of the theory of many sorted algebras. The concept of the signature of a many sorted algebra is introduced as well as the concept of many sorted algebra itself. Some auxiliary related notions are defined. The correspondence between ..."
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Cited by 125 (14 self)
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Summary. The basic purpose of the paper is to prepare preliminaries of the theory of many sorted algebras. The concept of the signature of a many sorted algebra is introduced as well as the concept of many sorted algebra itself. Some auxiliary related notions are defined. The correspondence between (1 sorted) universal algebras [8] and many sorted algebras with one sort only is described by introducing two functors mapping one into the other. The construction is done this way that the composition of both functors is the identity on universal algebras.
Directed sets, nets, ideals, filters, and maps
 Journal of Formalized Mathematics
, 1996
"... Summary. Notation and facts necessary to start with the formalization of continuous lattices according to [8] are introduced. The article contains among other things, the definition of directed and filtered subsets of a poset (see 1.1 in [8, p. 2]), the definition of nets on the poset (see 1.2 in [8 ..."
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Cited by 111 (30 self)
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Summary. Notation and facts necessary to start with the formalization of continuous lattices according to [8] are introduced. The article contains among other things, the definition of directed and filtered subsets of a poset (see 1.1 in [8, p. 2]), the definition of nets on the poset (see 1.2 in [8, p. 2]), the definition of ideals and filters and the definition of maps preserving arbitrary and directed sups and arbitrary and filtered infs (1.9 also in [8, p. 4]). The concepts of semilattices, supsemiletices and poset lattices (1.8 in [8, p. 4]) are also introduced. A number of facts concerning the above notion and including remarks 1.4, 1.5, and 1.10 from [8, pp. 3–5] is presented.
Boolean posets, posets under inclusion and products of relational structures
 Journal of Formalized Mathematics
, 1996
"... Summary. In the paper some notions useful in formalization of [11] are introduced, e.g. the definition of the poset of subsets of a set with inclusion as an ordering relation. Using the theory of many sorted sets authors formulate the definition of product of relational structures. ..."
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Cited by 86 (17 self)
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Summary. In the paper some notions useful in formalization of [11] are introduced, e.g. the definition of the poset of subsets of a set with inclusion as an ordering relation. Using the theory of many sorted sets authors formulate the definition of product of relational structures.
On Equivalents of Wellfoundedness  An experiment in Mizar
, 1998
"... Four statements equivalent to wellfoundedness (wellfounded induction, existence of recursively defined functions, uniqueness of recursively defined functions, and absence of descending omegachains) have been proved in Mizar and the proofs mechanically checked for correctness. It seems not to be w ..."
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Cited by 13 (3 self)
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Four statements equivalent to wellfoundedness (wellfounded induction, existence of recursively defined functions, uniqueness of recursively defined functions, and absence of descending omegachains) have been proved in Mizar and the proofs mechanically checked for correctness. It seems not to be widely known that the existence (without the uniqueness assumption) of recursively defined functions implies wellfoundedness. In the proof we used regular cardinals, a fairly advanced notion of set theory. The theory of cardinals in Mizar was developed earlier by G. Bancerek. With the current state of the Mizar system, the proofs turned out to be an exercise with only minor additions at the fundamental level. We would like to stress the importance of a systematic development of a mechanized data base for mathematics in the spirit of the QED Project.
On the Composition of Macro Instructions of Standard Computers
, 2000
"... this paper. 1. Preliminaries We use the following convention: k, m are natural numbers, x, X are sets, and N is a set with non empty elements ..."
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Cited by 4 (4 self)
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this paper. 1. Preliminaries We use the following convention: k, m are natural numbers, x, X are sets, and N is a set with non empty elements