Cyclic Ordering through Partial Orders *

International audienceThe orientation problem for ternary cyclic order relations has been attacked in the literature from combinatorial perspectives, through rotations , and by connection with Petri nets. We propose here a twofold characterization of orientable cyclic orders in terms of symmetries of partial orders as well as in terms of separating sets (cuts). The results are inspired by properties of non-sequential discrete processeses, but also apply to dense structures of any cardinality

Voting on Cyclic Orders, Group Theory, and Ballots

A cyclic order may be thought of informally as a way to seat people around a table, perhaps for a game of chance or for dinner. Given a set of agents such as $\{A,B,C\}$, we can formalize this by defining a cyclic order as a permutation or linear order on this finite set, under the equivalence relation where $A\succ B\succ C$ is identified with both $B\succ C\succ A$ and $C\succ A\succ B$. As with other collections of sets with some structure, we might want to aggregate preferences of a (possibly different) set of voters on the set of possible ways to choose a cyclic order. However, given the combinatorial explosion of the number of full rankings of cyclic orders, one may not wish to use the usual voting machinery. This raises the question of what sort of ballots may be appropriate; a single cyclic order, a set of them, or some other ballot type? Further, there is a natural action of the group of permutations on the set of agents. A reasonable requirement for a choice procedure would be to respect this symmetry (the equivalent of neutrality in normal voting theory). In this paper we will exploit the representation theory of the symmetric group to analyze several natural types of ballots for voting on cyclic orders, and points-based procedures using such ballots. We provide a full characterization of such procedures for two quite different ballot types for $n=4$, along with the most important observations for $n=5$.Comment: 29 pages, to be published in conference proceedings from AMS Special Session on The Mathematics of Decisions, Elections and Games, 202

Games on partial orders and other relational structures

This thesis makes a contribution to the classification of certain specific relational structures under the relation of n-equivalence, where this means that Player II has a winning strategy in the n-move Ehrenfeucht-Fraı̈ssé game played on the two structures. This provides a finer classification of structures than elementary equivalence, since two structures A and B are elementarily equivalent if and only if they are n-equivalent for all n. On each move of such a game, Player I picks a member of either A or B, and Player II responds with a member of the other structure. Player II wins the game if the map thereby produced from a substructure of A to a substructure of B is an isomorphism of induced substructures. Certain ordered structures have been studied from this point of view in papers by Mostowski and Tarski, for ordinals [22], and Mwesigye and Truss, for ordinals [25], some scattered orders, and finite coloured linear orders [24]. Here we extend the known results on linear orders by classifying them all up to 3-equivalence (which had previously been done for 2-equivalence), of which there are 281, using the method of characters. We also classify all partial orders up to 2-equivalence (there are 39), and discuss the difficulties of extending this to 3-equivalence, since the method of characters is not as effective as in the linear case. We classify (total) circular orders up to 3-equivalence, and relate the classification of partial circular orders to both these and to partial orders. A variety of related structures are discussed: trees, directed and undirected graphs, and unars (sets with a single unary function), which we categorise up to 2-equivalence. In a pebble game, the players of an otherwise standard Ehrenfeucht-Fraı̈ssé game are in addition provided with two identical sets of k distinguishable pebbles, and on each move they place a pebble on their chosen point. On each move, Player I may choose either to move a pebble to another point, or else use a new pebble, if any remain, and Player II must place the corresponding partner pebble. Such games correspond to logics in which there are only k variables, and moving a pebble corresponds to reusing the variable. Here we extend some work of Immerman and Kozen [14] on pebble games played on linear orders

Cyclic Ordering through Partial Orders

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