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In the context of partially ordered sets, why are the words for supremum and infimum "join" and "meet"? I find the nomenclature puzzling, especially since the English words "join" and "meet" are synonyms, but denote opposite concepts when talking about posets.

Does anyone know how these concepts got these names?

(Cross-posted on MSE)

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    $\begingroup$ When I join two things together, I combine them. This is analogous to "union". Where two things meet is what they have in common. This is analogous to "intersection". But this is only speculation. Another speculation: these words are translated from another language, such as German. $\endgroup$ Jul 5, 2019 at 11:12
  • $\begingroup$ @GeraldEdgar The mathematics post also has some similar speculation (though not about suspected German origin). I actually think of the "join" between two objects as the point where they touch, i.e. where they intersect. For example, the hip is the join between the leg and the torso (or the legs join the torso at the hip). But, one could also say the the legs meet the torso at the hip. $\endgroup$ Jul 5, 2019 at 11:16
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    $\begingroup$ In his book Lattice Theory, Birkhoff says that lattice ("Dualgruppe") was studied by Dedekind, 1897. If the terms come from German, that would be the place to look. $\endgroup$ Jul 5, 2019 at 11:21
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    $\begingroup$ @GeraldEdgar Birkhoff uses meets and joins already in his 1933 paper on lattices, where he rediscovered Dedekind's results before learning about them. So they did not come from Dedekind. I also checked Peirce, Menger and Klein, see below. It appears that it was Birkhoff's idea. $\endgroup$
    – Conifold
    Jul 6, 2019 at 11:17

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The "Elements" of modern lattice theory, from which the terminology spread is Birkhoff's Lattice Theory (1940). After defining lattices and introducing cups and caps (rather than wedges and vees) for inf and sup he says the following on the subject of meets and joins (p.17):

"A large fraction of the most important partly ordered systems considered in mathematics are lattices. Moreover in these systems, the opera tions $\frown$ and $\smile$ usually correspond to familiar and significant constructions.

Example 1. Let $\Sigma$ consist of all the subsets of any aggregate $I$, and let inclusion mean set-inclusion. Then "joins" are sums of sets and "meets" are set-products...

Example 3. Let $\Sigma$ consist of the subgroups of any group, and let inclusion mean set-inclusion. Then the terms "join" and "meet" have their usual meaning (also called union and intersection).

In other words, "join" joins (unites) subgroups together, and "meet" is where they meet (intersect). It is curious that Birkhoff considers the "usual meaning" to be for subgroups, not for sets. He does not credit anybody for the names, although he does mention that the definition is due to Peirce, and that Whitney suggested "cap" and "cup" names (Whitney himself used them for chains and cochains in homology).

The earliest appearance of meets and joins I found is in Birkhoff's first paper on lattices, On the combination of subalgebras (1933), where they are applied to subalgebras, subgroups and subrings, and again not credited to anybody. Birkhoff wrote this paper before he became aware of Dedekind's work. Von Neumann picked them up from Birkhoff in his later 1930-s papers. They do not appear in Peirce's On the Algebra of Logic (1885), or in Klein's expository paper Einige distributive Systeme in Mathematik und Logik (1929) that mentions many examples and names like conjuction/disjunction or union/intersection. Menger uses Summe and Durchschnitt (average) in Bemerkungen zu Grundlagenfragen (1928), but his 1936 New Foundations of Projective and Affine Geometry already has joins. I am going to speculate that Birkhoff was the terms' originator.

The beginnings of lattice theory go back to Boole, Dedekind and Peirce, but they did not have much of a follow-up until Emmy Noether revived Dedekind's work in her reshaping of algebra. Bilova in Lattice Theory – Its Birth and Life says that Birkhoff himself came up with the term "lattice" (under the influence of Hasse's drawings):

"Lattice structures started to be studied again at the end of 1920’s – this time in still another area of mathematics. Karl Menger presented the set of axioms characterizing projective geometries which are in fact complemented modular lattices. His investigations did not attract immediate attention, however, lattice structures soon appeared also in the field of formal logic (Fritz Klein who gave lattices its German name: “Verband”) and mainly algebra (Robert Remak, Oystein Ore). The biggest merits in the early developments of lattice theory belong to Garrett Birkhoff who also approached it from the side of algebra and united its various applications. In his first article about lattice structures [4] he rediscovered, apart from others, Dedekind’s results, and only after its publishing it was revealed that the studies of dual groups are identical with Birkhoff’s approach. G. Birkhoff also introduced the English word “lattice”, which is not the translation of its German equivalent, but was inspired by the image of some Hasse diagrams presenting lattices."

The [4] is the aforementioned 1933 paper. I should also mention that aside from lattice theory meet and join are now commonly used in modern geometric algebra/exterior calculus (not to be confused with van der Waerden's term for Euclid's book II), where they are applied to subspaces and Grassmann's multivectors. This was Rota's contribution, see his with co-authors On the exterior calculus of invariant theory, and The Many Lives of Lattice Theory.

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    $\begingroup$ Isn't lattice = Gitter in German? I am not a mathematician, but in chem. the German equivalent of lattice is Gitter. $\endgroup$
    – AChem
    Jul 5, 2019 at 12:32
  • $\begingroup$ @M.Farooq That would be the literal translation, apparently Birkhoff did not like Verband (association). $\endgroup$
    – Conifold
    Jul 5, 2019 at 12:37
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I did a quick survey on the origins of join and meet on Google books. Join and meet do not have German origins. See some early references. Ref: Information and Control 31, 312-340 (1976), Some Properties of Fuzzy Sets of Type 2 Masaharu Mizumoto & Kokichi Tanaka

join and meet

Another work rationalizes this terminology by Jerry M. Mendel and Robert I. Bob John (2002) enter image description here enter image description here

Edit: Confifold has provided early usages of meet and join.

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I do not have a citation for the etymology of the terms unfortunately. However, I know the following mnemonic. It's conceivable to me that the following was Birkhoff's reasoning (I have no evidence). If it helps direct anyone to find better evidence in Birkhoff's writings, that's all the better. If not, I think it's still a better way to explain the terminology to students.

Consider the projective plane. Define a set $L$ containing the empty set, the points of the plane, the lines of the plane, and the entire space. Equip $L$ with a partial order s.t. a point $p$ is less than a line $l$ if $p$ lies on $l$, and s.t. the empty set is less than everything and the entire plane greater than everything. Then $L$ is a lattice where for two distinct lines $l_0 \wedge l_1$ is the unique point at which the lines meet, and where for two distinct points $p_0 \vee p_1$ is the unique line joining the two points.

This construction may be generalized to a Grassmannian algebra (linear subspaces of $\mathbb{R}^n$) s.t. $S_0 \wedge S_1$ is the intersection of two linear subspaces (also a linear subspace) and $S_0 \vee S_1$ is the span of two linear subspaces (also a linear subspace). The projective plane as described above is isomorphic to the lattice of linear subspaces of $\mathbb{R}^3$ via the usual homogeneous coordinates for points in projective space.

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    $\begingroup$ Note that the join is not a union. Consequently the lattice of subspaces has particular pedagogical value in justifying lattice theory beyond the usual powerset and boolean algebra examples. $\endgroup$ Mar 19, 2022 at 4:14

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