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Concept

Relation

An equation a presentation imposes on words in its generators, saying that some product is the identity. Adding one is what distinguishes a quotient from a free object, and in topology it is what gluing a disc along a loop does to a fundamental group.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

j is one more than i, counting round — as a grid, with both quantifier readings. A grid of marks for a relation, with the row and column facts the two quantifier orders ask about.

Every row, or one column

For every person there is someone who loves them, and there is someone who loves everyone, are the same six words in a different order. Draw the relation as a grid and they become two obviously different questions — one about rows, one about columns.

logic · Quantifiers Three spaces, cut into pieces. 3 panels, drawn from: a sphere split into two caps meeting along a circle, two circles joined at a point, and a square whose opposite edges are glued into a torus. Each carries the fundamental group that van Kampen's theorem computes for it.

Cutting a space to find its group

A space assembled from two pieces has a fundamental group assembled from theirs, and the recipe is exact — take everything both groups offer and impose the relations the overlap forces. Almost every fundamental group anybody knows is computed this way, including all of the surfaces.

topology · Homotopy Six sentences from two quantifiers, and which imply which. A diagram of the six sentences that can be built from two quantifiers and a relation, arranged from strongest to weakest with arrows for implication, each labelled with how many of the 512 relations on three points satisfy it.

Six sentences from two quantifiers

One relation, two variables, 'for every' and 'there is': there are eight ways to arrange them and six different sentences come out. Which of them imply which is a small, complete diagram, found by checking all 512 relations on three points — and the diagram crosses over in the middle, which is where every confusion about the order of quantifiers lives.

logic · Quantifiers

Named alongside it

The objects these essays reach for when they reach for this one.

NegationPredicate logicQuantifierQuantifier orderCounterexampleDualityExhaustive searchFree groupFundamental groupGeneratorGluingGroup presentation

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