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ON REGULAR ANTICONGRUENCE IN ANTIORDERED SEMIGROUPS
"... Abstract. For an anticongruence q we say that it is regular anticongruence on semigroup (S,=, =, ·, α) ordered under antiorder α if there exists an antiorder θ on S/q such that the natural epimorphism is a reverse isotone homomorphism of semigroups. Anticongruence q is regular if there exists ..."
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Abstract. For an anticongruence q we say that it is regular anticongruence on semigroup (S,=, =, ·, α) ordered under antiorder α if there exists an antiorder θ on S/q such that the natural epimorphism is a reverse isotone homomorphism of semigroups. Anticongruence q is regular if there exists a quasiantiorder σ on S under α such that q = σ ∪ σ−1. Besides, for regular anticongruence q on S, a construction of the maximal quasiantiorder relation under α with respect to q is shown. 1. Introduction and
Constructive Suprema 1
"... Abstract: Partially ordered sets are investigated from the point of view of Bishop’s constructive mathematics, which can be viewed as the constructive core of mathematics and whose theorems can be translated into many formal systems of computable mathematics. The relationship between two classically ..."
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Abstract: Partially ordered sets are investigated from the point of view of Bishop’s constructive mathematics, which can be viewed as the constructive core of mathematics and whose theorems can be translated into many formal systems of computable mathematics. The relationship between two classically equivalent notions of supremum is examined in detail. Whereas the classical least upper bound is based on the negative concept of partial order, the other supremum is based on the positive notion of excess relation. Equivalent conditions of existence are obtained for both suprema in the general case of a partially ordered set; other equivalent conditions are obtained for subsets of a lattice and, in particular, for subsets of R n.
Suprema in ordered vector spaces: a constructive approach
"... Ordered vector spaces are examined from the point of view of Bishop’s constructive mathematics, which can be viewed as the constructive core of mathematics. Two different (but classically equivalent) notions of supremum are investigated in order to illustrate some features of constructive mathematic ..."
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Ordered vector spaces are examined from the point of view of Bishop’s constructive mathematics, which can be viewed as the constructive core of mathematics. Two different (but classically equivalent) notions of supremum are investigated in order to illustrate some features of constructive mathematics. By using appropriate definitions of the partial order set, supremum, and ordered vector space, one can prove constructively the usual properties of suprema of subsets of an ordered vector space. Furthermore, the paper provides constructive proofs of the usual properties of the modulus of a vector, proofs that avoid the dichotomy principle, a direct consequence of the law of excluded middle, a law of the classical logic which is viewed as the main source of nonconstructivism.