Meet and join: Difference between revisions

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== Mathematical preliminaries: join and meet of subgroups ==

In general, given some group G, the subgroups of G form an order-theoretic structure called a [https://en.wikipedia.org/wiki/Lattice_of_subgroups lattice of subgroups], where here "lattice" means [https://en.wikipedia.org/wiki/Lattice_(order) lattice in the order theory sense]; "~~trellis~~" in French, "Verband" in German. A lattice is a partially ordered set in which for two subgroups A and B of group G, we have A ≤ B iff A is itself a subgroup of B.

In general, given some group G, the subgroups of G form an order-theoretic structure called a [https://en.wikipedia.org/wiki/Lattice_of_subgroups lattice of subgroups], where here "lattice" means [https://en.wikipedia.org/wiki/Lattice_(order) lattice in the order theory sense]; "treillis" in French, "Verband" in German. A lattice is a partially ordered set in which for two subgroups A and B of group G, we have A ≤ B iff A is itself a subgroup of B.

Given two subgroups A and B, the '''join''' of A and B is the smallest subgroup of G containing both; this is sometimes also called the '''subgroup generated by A and B.''' The '''meet''' of A and B is the intersection of both.

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 == Mathematical preliminaries: join and meet of subgroups ==
-In general, given some group G, the subgroups of G form an order-theoretic structure called a [https://en.wikipedia.org/wiki/Lattice_of_subgroups lattice of subgroups], where here "lattice" means [https://en.wikipedia.org/wiki/Lattice_(order) lattice in the order theory sense]; "trellis" in French, "Verband" in German. A lattice is a partially ordered set in which for two subgroups A and B of group G, we have A ≤ B iff A is itself a subgroup of B.
+In general, given some group G, the subgroups of G form an order-theoretic structure called a [https://en.wikipedia.org/wiki/Lattice_of_subgroups lattice of subgroups], where here "lattice" means [https://en.wikipedia.org/wiki/Lattice_(order) lattice in the order theory sense]; "treillis" in French, "Verband" in German. A lattice is a partially ordered set in which for two subgroups A and B of group G, we have A ≤ B iff A is itself a subgroup of B.
 Given two subgroups A and B, the '''join''' of A and B is the smallest subgroup of G containing both; this is sometimes also called the '''subgroup generated by A and B.''' The '''meet''' of A and B is the intersection of both.