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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 38 (4 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 4 (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.
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
 Add to MetaCart
“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