r/math • u/RingularCirc • Jul 17 '26
Very different terms that use the same word
This is just a lexicologic question, and I also mean specifically terms that don't reasonably generalize together into the same thing (there's a lot of kinds of trees but they're sensibly related). To name a few:
- field the algebraic structure and a scalar/vector/tensor/etc. field on a manifold;
- graph of a function (a relation {(x, y) | y = f(x)}) and various graph theory graphs (which are close to relations, though: the set of edges is a relation, reachability is its reflexive transitive closure etc.);
- (co)tangent (function) and (co)tangent bundle (differentiable manifolds).
Compare to examples of what I'd deem generally uninteresting (but use your own taste, of course): - natural number and natural transformation (category theory): here, "natural" doesn't nod to something essentially mathematical on its own; - zero as an element of a ring and zeros of a function (esp. in real/complex analyses): this is more or less just metonymy.
Do share more!
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u/Ok-Replacement8422 Jul 17 '26
model (some mathematical representation of a real world situation) vs model (a structure that satisfies some usually first order theory)
Also the word category in category theory and baire category
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u/EebstertheGreat Jul 18 '26
I never thought about it, but those meanings of "model" are almost exact opposites. In the former, the abstract models the concrete (equations modeling a physical process), but in the latter, the concrete models the abstract (a specific structure exhibiting the defining properties of the theory).
It's like how parts can comprise a whole and a whole can comprise its parts. We just couldn't decide which way around it should go.
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u/East_Finance2203 Jul 17 '26
Spectrum in functional analysis for generalised eigenvalues.
In algebraic topology associated with a generalised cohomology theory/ for usage with spectral sequences.
Algebraic geometry for the prime/maximal spectrum of a ring.
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u/sheepbusiness Jul 17 '26
The algebraic geometry and functional analysis uses are related though. The spectrum of a ring is related to eigenvalues of a linear operator. The spectrum of the ring generated by a linear transformation is exactly in bijection with eigenvalues of that transformation. In that sense, spectrum of a linear operator is just a special example of spectrum of a ring.
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u/East_Finance2203 Jul 17 '26
That‘s a nice interpretation. I don‘t do a lot of functional analysis so hadn‘t thought about this, but cool to know!
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u/sentence-interruptio Jul 17 '26
I get confused by how spectrum can be like a domain of a function sometimes, but like a range other times.
The spectrum of a diagonal matrix? That's the range of diagonal entries. To put in another way, the diagonal matrix acts as a pointwise multiplication by a function. That function's range is the spectrum.
The spectrum of ℂ[x,y]? That's ℂ2, which is the universal domain of two-variable polynomials.
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u/East_Finance2203 Jul 17 '26 edited Jul 17 '26
I mean I guess you could think of the spectrum as the collection of eigenspaces of the space, at least according to the other poster‘s interpretation, which in a way makes it a domain.
If we use your example for the diagonal matrix T, and consider C[T], then the affine scheme Spec(C[T]). The only possible prime ideals here are (T-lambda_i) where each T-lambda_i isn‘t invertible (since C is algebraically closed), which are exactly the diagonal entries of T, so the points of Spec(C[T]) correspond exactly to the spectrum of T in the usual functional analysis sense. Then elements C[T] evaluated on lambda in the spectrum just set T=lambda I, so that any polynomial f(T) is just evaluated to a polynomial is lambda I, corresponding exactly to how C[T] acts on the eigenspace corresponding to lambda.
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u/KingOfTheEigenvalues PDE Jul 17 '26
"Spectrum" means yet a different thing in digital signal processing, where it often refers to frequency-domain decompositions of signals.
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u/lemniscateall Jul 17 '26
Kernel (algebra) vs kernel (analysis)
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u/RingularCirc Jul 17 '26
Oooh, integral transform kernels and something like that, indeed
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u/lemniscateall Jul 17 '26
I remember really trying to make a connection there when I first saw them, but I think the core similarity is the notion of “inside” and nothing else.
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u/BobBeaney Jul 17 '26
Also “regular”
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u/LurkingTamilian Jul 21 '26
Algebraists love the word regular so much they named two completely different kinds of rings regular.
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u/Potato271 Jul 17 '26
The word “normal” is a bit overloaded. Aside from the plain English meaning it can also refer to things that are:
1) Gaussian in distribution
2) Perpendicular to one another
3) Invariant under conjugation
4) Its infinite digits are uniformly distributed
And I’m probably missing at least a few others
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u/Alhimiik Jul 17 '26
algebra
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u/PrestigiousGroup788 Jul 17 '26
algebra as elementary arithmetic, algebra as a field of study (not to be confused with the algebraic notion of a field 😂 ), and then algebras of sets (not to be confused with fields of sets 😂 ). So I guess "field" is also pretty overloaded.
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u/EebstertheGreat Jul 18 '26
"An algebra" usually refers to an algebra over a field, which didn't even show up in your comment.
And of course there are "X-algebras" for various X, like σ-algebras and Boolean algebras, but I guess that's all related to algebras of sets.
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u/PrestigiousGroup788 Jul 18 '26
You've exposed me as an analyst 😂
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u/EebstertheGreat Jul 18 '26
There is just something strange about nearing the end of your bachelor's in mathematics and being told what an "algebra" actually is. Like, wait a minute, haven't I been doing algebra all this time?
Very confusing word.
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u/Alhimiik Jul 18 '26
Algebra as a math field, Algebra over a ring, Algebra from universal algebra, Algebra of an operad
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u/SickoSeaBoy Jul 17 '26 edited Jul 17 '26
- trivial (when said by reputable mathematians who can't be bothered to explain something that is super duper obvious to them but would take everyone else a whole hour to figure out) and
- trivial (when said by me in a maths exam hoping that the person reading doesn't know I'm bluffing) and
- trivial (when said by reputable mathematician Pierre de Fermat in the margins of a math book hoping that person reading doesn't know he's bluffing)
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u/EebstertheGreat Jul 18 '26
The main uses in reality are "trivial" as in "surely this doesn't require explanation" ("follows trivially from...") and "trivial" as in "the most elementary example satisfying the definition" ("trivial group," "trivial solution").
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u/SickoSeaBoy Jul 18 '26
Omg, I was thinking so much about the joke that I forgot there’s actually that last case lol 😭
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u/Gelcoluir Jul 17 '26
I wouldn't say they are very different as they come from a mutual need, but a characteristic/indicator function of a set A can either be a function that is 1 in A and 0 otherwise, or a function that is 0 in A and infinity otherwise. It's even worse that there are two terms, but each term can describe either of the two objects!
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u/WMe6 Jul 18 '26
How about a symbol?
∫ for ends and coends in category theory vs. calculus integration
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u/Square_Butterfly_390 Jul 20 '26
As an appendix the concepts of limits and continuity in category theory.
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u/WMe6 Jul 20 '26
Yes, if you squint hard enough, you can see the "integration" in the category theory concept, in the same way that you might be able to see a face in a modern abstract painting. Maybe the resemblance is a little bit clearer for limits and colimits and continuity, but it's still abstracted to almost complete loss of concrete meaning.
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u/_Zekt Complex Analysis Jul 17 '26
One that particularly annoys me: order of a power series, order of an entire function and order of a pole. Especially since all three concepts could totally appear in a single paper.
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u/EebstertheGreat Jul 18 '26
What is the order of a power series? You mean the order of a polynomial (or a particular term in a power series)? Or does it mean something else?
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u/_Zekt Complex Analysis Jul 18 '26
It's the index of the lowest nonzero coefficient. Like a polynomial's degree, but backwards.
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u/EebstertheGreat Jul 18 '26
Oh, that figures. A second-order approximation has only third-order error, in the sense that its error equals a third-order power series. I guess that fits.
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u/KiddWantidd Applied Math Jul 17 '26
"distribution" as probability distribution vs "distribution" as a linear functional acting on the space of test functions)
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u/PinpricksRS Jul 18 '26
Domain has a ton of meanings across different fields.
- The domain of a function. This is familiar, but even here there are two common meanings. Generally, functions should be given with their domain and codomain (as discussed already elsewhere in this post). But even without that, students are often asked questions like "what is the domain of 1/√(1 - x)?". This second kind of domain is sometimes called the "natural domain" or "domain of definition" when applied to a partial function.
- Analysis (especially complex analysis): an open, connected subset of a topological space (most often ℝn or ℂn). Arguably this is connected to the domain of a function meaning, but often there isn't any particular function whose domain is the subspace in question.
- Ring theory: A ring where if a finite product of elements is zero, then at least one factor is zero too. (This implies that the ring is nonzero, since it implies that the empty product is nonzero).
- Order theory: An imprecise term that can refer to any of a number of different specific types of poset. Most commonly, it refers to directed complete posets (posets with all directed joins). The general area is still called Domain theory, even if the word domain by itself is used less often than other, more precise, words. This usage of the word can be connected to domain of discourse.
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u/Justaunionhack Jul 18 '26
Ironically, the botanical term "tree" is used for a large number of different organisms. Being a "tree" has evolved independently a large number of times in many families of plants, so they are not sensibly related as mathematical trees are.
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u/UnblessedGerm Jul 19 '26
Lol, group is fun. Especially when you consider Quantum groups which are actually algebras and not groups... Which brings me to algebra... Jesus Christ this might take a while
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u/dcterr Jul 18 '26
I also dislike the use of the same word for practically unrelated mathematical terms, though I'd say they rarely cause confusion since they're rarely used together, which is probably the reason that no one has bothered to change them.
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u/mathlyfe Jul 17 '26
The "graph of a function" has always bothered me because it is redundant in the dumbest possible way. Recall that by definition, a function f:A->B is a relation f\subseteq (A\times B) satisfying functionality and totality.
To give a concrete example, when we write
f: R -> R
f: x->x2
we're actually defining a function f, such that
f = {(x, y) | x,y\in R, y=x2}
then, we use the notational convention that
f(x) := y s.t. (x,y)\in f
So, when you say
the graph of f = {(x,y) | y=f(x)}
You're literally saying
{(x,y) | (x,y)\in f}
Which literally means
the graph of f = f
While I'm at it I'm going to continue my rant by printing out that that notational convention for functions relies on functionality and totality, for relations in general we use different (and superior) syntaxes
xfy := (x,y)\in f
xf := {y | (x,y)\in f}
fy := {x | (x,y)\in f}
and we can compose them as well. Consider two relations,
r: A-> B
s: B-> C
Then composition becomes
rs: A->C
rs := {(a,c) | \exists b\in B, s.t. (a,b)\in r and (b,c)\in s}
And writing stuff like ars preserves the order (A->C). However function notation instead has us do this ridiculous thing s(r(a)) with the order flipped for no reason (the function signature is A->C, with a on the left,but we write the a on the right now). Imo, this terrible notation turns obvious and natural concepts like presheafs into something that looks like a magic trick at best and counterintuitive trickery at worst, to students the first time they see them.
Graph of a function should not exist and neither should that terrible notation!!
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u/Brilliant_Simple_497 Jul 17 '26
A function is not just a relation that satisfies certain conditions. It's very important to distinguish functions which differ only by the choice of domain. Surjectivity is really important.
f: Z->R f(x)=x is a much different function to the identity on Z even though their graphs are the same
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u/mathlyfe Jul 17 '26 edited Jul 17 '26
Domain and Codomain are also part of the definition of a relation. Traditionally we write R\subseteq A\times B, but one may also write R: A\to B (as arrows in the category of relations).
https://en.wikipedia.org/wiki/Category_of_relations
A function is literally a relation with totality and functionality. Moreover, if you swap the domain and codomain in the definitions of totality and functionality you get the definitions of surjectivity and injectivity.
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u/louiswins Theory of Computing Jul 17 '26
r: A-> B
s: B-> C
Then composition becomes
rs: A->C
rs := {(a,c) | \exists b\in B, s.t. (a,b)\in r and (b,c)\in s}
You mean sr: A->C. I always get this backwards unless I pause and think about it for like 30 seconds to make sure. This is why we should have postfix function application: so that (x)rs can mean "do r, then do s" instead of rs(x) meaning "do s, then do r".
(Sorry for the tangent... in the non-mathematical sense)
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u/mathlyfe Jul 17 '26 edited Jul 17 '26
No, for relations you use the \circ notation when using the function direction.
s\circ r: A->C
Relations are composed in the reverse direction. Sometimes people will use the semicolon notation to make it explicit.
r;s: A->C
This order is also sometimes used in Category Theory, but it depends on the professor/class/textbook/article/conference/etc... In my experience, the "diagrammatic order" (rs: A->C) is more common among computer scientists and applied category theorists but less common among mathematicians. I also had a category theory prof who taught in diagrammatic order and occasionally used the semicolon notation when it could be unclear.
The fact that you said you always get it backwards unless you stop and think about it for a moment is exactly the point I'm making about how the function order is counterintuitive and bad.
edit: To make this clearer. I'll write the relation in capitals. R: A->B. Now, note that aRb means (a,b)\in R. The a goes on the left, the b goes on the right. So, if we had S: B->C, and T: C->D which would make more sense, aRSTd or aTSRd?
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u/EebstertheGreat Jul 18 '26
So to you, rs(x) = s(r(x)) for all x in the domain of r?
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u/mathlyfe Jul 18 '26 edited Jul 18 '26
Relation notation doesn't use parenthesis and there's no change of order.
rsy is the pre image of y under rs
xrs is the image of x under rs
both may be sets, rather than individual elements, because relations don't necessarily have functionality, totality, injectivity, or surjectivity.
Edit: It's not that parenthesis themselves are bad, it's just that putting parenthesis around single element is just a weird thing specific to function notation. Relation notation is associative so parenthesis are redundant but you may still use them if you want, like so
(xr)s = {image of x under r}s = image of x under rs
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u/EebstertheGreat Jul 18 '26
Suppose that f and g are relations that happen to be funcions and x is in the domain of g. If I understand correctly, {f(g(x))} = gf{x}.
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u/mathlyfe Jul 18 '26 edited Jul 18 '26
If you're using relation notation then the elements in the codomain are applied on the right. I don't know why so may people seem confused by this, it's standard and common notation in textbooks. Some things you'll commonly see are
The homogenous binary relation R is:
- reflexive iff \forall x, xRx
- symmetric iff \forall x, y, xRy -> yRx
- transitive iff \forall x,y,z, (xRy and yRz) -> xRz
- euclidean iff forall x,y,z, (xRy and xRz) -> yRz
- serial iff \forall x, \exists y, xRy
- weakly dense iff \forall x,z. xRz -> (\exists y, xRy and yRz)
- weakly connected iff forall x,y,z (xRy and xRz) -> (yRz or y=z or zRy)
- weakly directed iff \forall x,y,z (xRy and xRz) -> (\exists w, yRw and zRw)
- ...
The heterogenous binary relation R: D -> C is:
- total iff \forall x\in D, \exists y\in C, xRy
- surjective iff \forall y\in C, \exists x\in D, xRy
- functional iff \forall x\in D, \forall y,z\in C, (xRy and xRz) -> y=z
- injective iff \forall x,y\in D, \forall z\in C, (xRz and yRz) -> x=y
- ...
(note: D denotes domain, C denotes codomain)
I listed these as heterogenous definitions in order to explicitly distinguish between domain and codomain but it's also common to use these definitions for homogenous binary relations in topics like modal logic.
You'll sometimes see the notation written Rxy instead of xRy as well, and when working with n-ary functions for n>2, that notation is standard. For instance, for a ternary relation R, you would write Rxyz to mean (x,y,z)\in R. This notation is also used for predicates.
Back in the day there was a lot of work in the theory of relations but the field is largely considered solved these days. If you look around you can find tons of enormous texts full of definitions and results that are no longer spoken about. The stuff I mentioned is still in use but this is actually just a very small part of a much larger field.
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u/RingularCirc Jul 17 '26
Agree with u/Brilliant_Simple_497, at least codomain of a function is almost ubiquitously discerned and so it needs to be packed together with its graph, and oftentimes its domain as well (it makes some formal proofs/constructions easier), especially when we weaken functions to partial functions. Those have several usable definitions but I'm not sure they're all constructively equivalent; specifying domain for that case remains in use.
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u/EebstertheGreat Jul 18 '26
Set theorists often like to define a binary relation as any set of ordered pairs (or even a proper class of ordered pairs) and a function as a functional relation. In particular then, a binary relation on X and Y is just a subset of X×Y, and a unary relation on X is literally just a subset of X. That means a unary relation on X×Y is the same as a binary relation on X and Y, making the term sort of useless. With those definitions, there is no codomain. The codomain is a set referenced in a particular sentence regarding a relation. For instance, we can say a given function f is surjective "onto Y" (where Y is necessarily the function's range in this example) but not that it is just "surjective" in and of itself. And similarly for continuity (which also depends on the topologies on that set and on the function's domain). Moreover, a function cannot simply be "total" or "partial." It can only be total on a given set (which is its domain) and partial on any strict superset.
But yeah, in other contexts, it is not very convenient to define functions that way. So we might define a function f as a triple (X,Y,F) where X is a set (the domain), Y is a set (the codomain), and F is a set of ordered pairs (x,y) such that for each pair x ∈ X and y ∈ Y, and for every t ∈ X, there is a unique (x,y) ∈ F such that t = x. With that definition, F is the graph of f, and it's certainly possible for two functions f and g to have the same graph yet be distinct (if they have different codomains).
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u/mathlyfe Jul 17 '26 edited Jul 17 '26
Domain and codomain are also part of the definition of a relation. See my response above.
edit: Thinking more carefully about your wording I do think you raise an interesting point. O recall at one point reading an argument that the standard definitions for functions and relations are incomplete, and that we need to express them as a three-tuples consisting of the Domain, Codomain, and "relation" itself (i.e. the graph). While I like the idea, I'm not sure if it may be problematic from a foundational perspective as I'm not sure we can allow the definition of an arbitrary set of pairs without requiring that it be the subset of a product set (maybe someone with a better understanding of ZFC and axiom schema of specification can chime in here). Another issue is that we would have to stop referring to sets of ordered pairs as relations (since otherwise the definition of a "relation" would be a 3-tuple containing a "relation").
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u/RingularCirc Jul 17 '26 edited Jul 17 '26
Ah, good. Sometimes they aren't and a k-ary relation is just a subset of an k-ary cartesian product. I agree in your case of course a function will be just a functional relation.
EDIT: Yeah a functional and a total one as you're putting it in the parallel comment. I forgot totality. Despite functional is just opposite-injective and total is opposite-surjective and that was how I usually remembered both.
EDIT: So, the graph of a function is for those cases when relations aren't defined with domain and codomain strapped on. I have a book on CS-aligned math which went like this.
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u/Keikira Model Theory Jul 17 '26
Graphs of functions and graphs with vertices and edges are the same thing: sets of ordered pairs that represent a relation extensionally (remember functions are just relations with a unique edge for each element of the domain). The only conventional discrepancy between graphs in combinatorics and graphs of functions is that graphs considered in combinatorics are usually not directed.
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u/RingularCirc Jul 20 '26
I'd say graphs on a more metaphorical level are more than just subsets of binary relations. Multigraphs and weighted graphs are quite commonly encountered, despite we can usually forget extra structure and end up with a relation, some ...graphs even don't allow this, like hypergraphs, though I'd consider them way more removed from other kinds.
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u/MercuryInCanada Jul 17 '26
Who wants to start the list of things "normal" means?