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Utility Representation of an Incomplete Preference Relation
 Journal of Economic Theory
, 2002
"... We consider the problem of representing a (possibly) incomplete preference relation by means of a vectorvalued utility function. Continuous and semicontinuous representation results are reported in the case of preference relations that are, in a sense, not “too incomplete. ” These results generaliz ..."
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Cited by 64 (6 self)
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We consider the problem of representing a (possibly) incomplete preference relation by means of a vectorvalued utility function. Continuous and semicontinuous representation results are reported in the case of preference relations that are, in a sense, not “too incomplete. ” These results generalize some of the classical utility representation theorems of the theory of individual choice, and paves the way towards developing a consumer theory that realistically allows individuals to exhibit some “indecisiveness ” on occasion.
ORDERINGS OF MONOMIAL IDEALS
, 2003
"... We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular ..."
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Cited by 5 (1 self)
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We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular, we give an interpretation of the height function in terms of the HilbertSamuel polynomial, and we compute upper and lower bounds on the maximal order type.
© Hindawi Publishing Corp. ARCHIMEDEAN UNITAL GROUPS WITH FINITE UNIT INTERVALS
, 2002
"... Let G be a unital group with a finite unit interval E,letn be the number of atoms in E,andletκ be the number of extreme points of the state space Ω(G). We introduce canonical orderpreserving group homomorphisms ξ: Z n → G and ρ: G → Z κ linking G with the simplicial groups Z n and Z κ. We show that ..."
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Let G be a unital group with a finite unit interval E,letn be the number of atoms in E,andletκ be the number of extreme points of the state space Ω(G). We introduce canonical orderpreserving group homomorphisms ξ: Z n → G and ρ: G → Z κ linking G with the simplicial groups Z n and Z κ. We show that ξ is a surjection and ρ is an injection if and only if G is torsionfree. We give an explicit construction of the universal group (unigroup) for E using the canonical surjection ξ. IfG is torsionfree, then the canonical injection ρ is used to show that G is Archimedean if and only if its positive cone is determined by a finite number of homogeneous linear inequalities with integer coefficients. 2000 Mathematics Subject Classification: 06F20. 1. Introduction and basic definitions. In this paper, we continue the study of unital groups with finite unit intervals begun in [2, 3]. Motivation for this study can be found in [2]. Although we will attempt to keep this paper somewhat selfcontained, we make free use of the notation, nomenclature, and results
CHAIN DECOMPOSITION THEOREMS FOR ORDERED SETS AND OTHER MUSINGS
, 1997
"... This paper is dedicated to the memory of Prof. Garrett Birkhoff Abstract. A brief introduction to the theory of ordered sets and lattice theory is given. To illustrate proof techniques in the theory of ordered sets, a generalization of a conjecture of Daykin and Daykin, concerning the structure of p ..."
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This paper is dedicated to the memory of Prof. Garrett Birkhoff Abstract. A brief introduction to the theory of ordered sets and lattice theory is given. To illustrate proof techniques in the theory of ordered sets, a generalization of a conjecture of Daykin and Daykin, concerning the structure of posets that can be partitioned into chains in a “strong ” way, is proved. The result is motivated by a conjecture of Graham’s concerning probability correlation inequalities for linear extensions of finite posets. 1.
Ordered Sets
"... “And just how far would you like to go in? ” he asked.... “Not too far but just far enough so’s we can say that we’ve been there, ” said the first chief. “All right, ” said Frank, “I’ll see what I can do.” –Bob Dylan In group theory, groups are defined algebraically as a model of permutations. The C ..."
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“And just how far would you like to go in? ” he asked.... “Not too far but just far enough so’s we can say that we’ve been there, ” said the first chief. “All right, ” said Frank, “I’ll see what I can do.” –Bob Dylan In group theory, groups are defined algebraically as a model of permutations. The Cayley representation theorem then shows that this model is “correct”: every group is isomorphic to a group of permutations. In the same way, we want to define a partial order to be an abstract model of set containment ⊆, and then we should prove a representation theorem for partially ordered sets in terms of containment. A partially ordered set, or more briefly just ordered set, is a system P = (P, ≤) where P is a nonempty set and ≤ is a binary relation on P satisfying, for all x,y,z ∈ P, (1) x ≤ x, (reflexivity) (2) if x ≤ y and y ≤ x, then x = y, (antisymmetry) (3) if x ≤ y and y ≤ z, then x ≤ z. (transitivity) The most natural example of an ordered set is P(X), the collection of all subsets of a set X, ordered by ⊆. Another familiar example is Sub G, all subgroups of a group G, again ordered by set containment. You can think of lots of examples of this type. Indeed, any nonempty collection Q of subsets of X, ordered by set containment, forms an ordered set. More generally, if P is an ordered set and Q ⊆ P, then the restriction of ≤ to Q is a partial order, leading to a new ordered set Q. The set ℜ of real numbers with its natural order is an example of a rather special type of partially ordered set, namely a totally ordered set, or chain. C is a chain if for every x,y ∈ C, either x ≤ y or y ≤ x. At the opposite extreme we have antichains, ordered sets in which ≤ coincides with the equality relation =. We say that x is covered by y in P, written x ≺ y, if x < y and there is no z ∈ P with x < z < y. It is clear that the covering relation determines the partial order in a finite ordered set P. In fact, the order ≤ is the smallest reflexive, transitive relation containing ≺. We can use this to define a Hasse diagram for a finite ordered set P: the elements of P are represented by points in the plane, and a line is drawn from a up to b precisely when a ≺ b. In fact this description is not precise, but it 1 (b) antichain (a) chain
1. Ordered Sets
"... “And just how far would you like to go in? ” he asked.... “Not too far but just far enough so’s we can say that we’ve been there, ” said the first chief. “All right, ” said Frank, “I’ll see what I can do.” –Bob Dylan In group theory, groups are defined algebraically as a model of permutations. The C ..."
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“And just how far would you like to go in? ” he asked.... “Not too far but just far enough so’s we can say that we’ve been there, ” said the first chief. “All right, ” said Frank, “I’ll see what I can do.” –Bob Dylan In group theory, groups are defined algebraically as a model of permutations. The Cayley representation theorem then shows that this model is “correct”: every group is isomorphic to a group of permutations. In the same way, we want to define a partial order to be an abstract model of set containment ⊆, and then we should prove a representation theorem for partially ordered sets in terms of containment. A partially ordered set, or more briefly just ordered set, is a system P = (P,≤) where P is a nonempty set and ≤ is a binary relation on P satisfying, for all x,y,z ∈ P, (1) x ≤ x, (reflexivity) (2) if x ≤ y and y ≤ x, then x = y, (antisymmetry) (3) if x ≤ y and y ≤ z, then x ≤ z. (transitivity) The most natural example of an ordered set is P(X), the collection of all subsets of a set X, ordered by ⊆. Another familiar example is Sub G, all subgroups of a group G, again ordered by set containment. You can think of lots of examples of this type. Indeed, any nonempty collection Q of subsets of X, ordered by set containment, forms an ordered set. More generally, if P is an ordered set and Q ⊆ P, then the restriction of ≤ to Q is a partial order, leading to a new ordered set Q. The set ℜ of real numbers with its natural order is an example of a rather special type of partially ordered set, namely a totally ordered set, or chain. C is a chain if for every x,y ∈ C, either x ≤ y or y ≤ x. At the opposite extreme we have antichains, ordered sets in which ≤ coincides with the equality relation =. We say that x is covered by y in P, written x ≺ y, if x < y and there is no z ∈ P with x < z < y. It is clear that the covering relation determines the partial order in a finite ordered set P. In fact, the order ≤ is the smallest reflexive, transitive relation containing ≺. We can use this to define a Hasse diagram for a finite ordered set P: the elements of P are represented by points in the plane, and a line is drawn from a up to b precisely when a ≺ b. In fact this description is not precise, but it 1 (b) antichain (a) chain
1. Ordered Sets
"... “And just how far would you like to go in? ” he asked.... “Not too far but just far enough so’s we can say that we’ve been there, ” said the first chief. “All right, ” said Frank, “I’ll see what I can do.” –Bob Dylan In group theory, groups are defined algebraically as a model of permutations. The C ..."
Abstract
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(Show Context)
“And just how far would you like to go in? ” he asked.... “Not too far but just far enough so’s we can say that we’ve been there, ” said the first chief. “All right, ” said Frank, “I’ll see what I can do.” –Bob Dylan In group theory, groups are defined algebraically as a model of permutations. The Cayley representation theorem then shows that this model is “correct”: every group is isomorphic to a group of permutations. In the same way, we want to define a partial order to be an abstract model of set containment ⊆, and then we should prove a representation theorem for partially ordered sets in terms of containment. A partially ordered set, or more briefly just ordered set, is a system P = (P,≤) where P is a nonempty set and ≤ is a binary relation on P satisfying, for all x, y, z ∈ P, (1) x ≤ x, (reflexivity) (2) if x ≤ y and y ≤ x, then x = y, (antisymmetry) (3) if x ≤ y and y ≤ z, then x ≤ z. (transitivity) The most natural example of an ordered set is P(X), the collection of all subsets of a set X, ordered by ⊆. Another familiar example is Sub G, all subgroups of a group G, again ordered by set containment. You can think of lots of examples of this type. Indeed, any nonempty collection Q of subsets of X, ordered by set containment, forms an ordered set. More generally, if P is an ordered set and Q ⊆ P, then the restriction of ≤ to Q is a partial order, leading to a new ordered set Q. The set ℜ of real numbers with its natural order is an example of a rather special type of partially ordered set, namely a totally ordered set, or chain. C is a chain if for every x, y ∈ C, either x ≤ y or y ≤ x. At the opposite extreme we have antichains, ordered sets in which ≤ coincides with the equality relation =. We say that x is covered by y in P, written x ≺ y, if x < y and there is no z ∈ P with x < z < y. It is clear that the covering relation determines the partial order in a finite ordered set P. In fact, the order ≤ is the smallest reflexive, transitive relation containing ≺. We can use this to define a Hasse diagram for a finite ordered set P: the elements of P are represented by points in the plane, and a line is drawn from a up to b precisely when a ≺ b. In fact this description is not precise, but it 1 (a) chain (b) antichain (c) P{x, y, z}