# What is an ideal-supporting algebra?

I'm sorry if this question is too elementary, but I asked it at MathStackExchange and it received no responses.

On the Wikipedia page for congruence relation it mentions how for groups and rings, congruences can be identified with normal subgroups and ideals respectively, and that the most general algebraic structure for which this can be done are ideal-supporting algebras. But I haven't been able to discover what an ideal-supporting algebra is.

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A soft-question here is not the opposite of a hard-question :) By contrast the soft-question tag is (only) to be used for (certain) questions that are not really mathematical questions. I thus removed it and added a top-level tag instead. – user9072 Feb 12 '13 at 19:07
I have not heard of "ideal-supporting algebra" either. I suspect it was invented for the article because the author could not remember a more appropriate term. (Neither can I.) I recommend looking more into congruences; there are permuting congruences, congruence modular algebras and varieties, Hamltonian algebras, and other properties that rely heavily on congruences. I think you will need an algebra with zero, without one, maybe some uniformity (Hamiltonian?), to approach this mystery of "ideal-supporting algebra" Gerhard "Forgotten More Than I Knew" Paseman, 2013.02.12 – Gerhard Paseman Feb 12 '13 at 19:38
@Gerhard Paseman: One should be flexible about the meaning of "zero" and "one" in your comment. In the variety of Heyting algebras, a congruence relation is determined by a filter (the congruence class containing the top element 1), but not by an ideal (the congruence class containing the bottom element 0). Of course, there's no problem here if you're willing to stand on your head (or to turn the Heyting algebra upside down). – Andreas Blass May 10 '13 at 14:47

It looks like the term ‘ideal-supporting algebra’ was written by me and survived slightly more than a decade on Wikipedia without being altered. (Well, somebody added a hyphen, a change that I agree with.) Since I put brackets around it, I'm sure that I must have heard the term somewhere, but I couldn't tell you now. Now that I think of it, a more precise term would be ‘ideal-supporting variety [of algebras]’.

And if I search for that phrase, I find it in Eric Schechter's 1996 Handbook of Analysis and its Foundations (which for some reason Google Books has classified under Business & Economics). Since I was reading this book a lot a decade ago, that's probably what I meant all along. Shechter often invented terminology for his book, when terminology in the literature was inconsistent or missing, so I wouldn't be surprised if it's essentially unique to him.

At this point, probably the best thing for me to do is to edit Wikipedia for a little bit, finishing what I started in 2002.

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Actually, rereading Schechter reminds me that the standard term in the literature is ‘Ω-group’, and Schechter invented ‘ideal-supporting variety’ to refer to the simplified situation when the Ω-group operation is commutative. And although Ω-groups aren't on Wikipedia, you can actually find out about them in the literature on universal algebra, so that may be all the answer you need. Or search for ‘ideal-supporting variety’ and read Schechter's account on Google Books. – Toby Bartels Feb 12 '13 at 21:07
Actually Ω-groups are on WP, but I didn't recognise them at first, since WP discusses only the particular (but original) case in which the non-group operations are all unary. (And since that doesn't include rings, no surprise that WP's article doesn't cover ideals.) The article on congruences now links properly. – Toby Bartels Mar 23 '13 at 5:02
What happened to the ideal-supporting algebra article on Wikipedia? I can't find it! – Stavros Macrakis Jan 10 at 22:17
An article with that name never existed on Wikipedia. Instead, the term was used in the article on congruence relations. Later I changed it to the more standard term Ω-group, so now you won't find the term on Wikipedia at all. The only Wikipedia article on the subject is en.wikipedia.org/wiki/Group_with_operators, but unfortunately that still only covers the case where the non-group operations are all unary. The more general case is covered on the nLab at ncatlab.org/nlab/show/Omega-group, but there is little more than a definition there. At least it has some references. – Toby Bartels Jan 22 at 18:44
Or for the commutative case, you can read Schechter, as I remarked above. Google is currently showing most of it to me at books.google.com/… (pages 221–227). – Toby Bartels Jan 22 at 18:52

Something for Toby Bartels as well as the poster, I've decided to post as an answer.

A (Universal) algebra A is Hamiltonian if for every subalgebra B of A there is a congruence of A in which B is a congruence class. This is a little stronger notion than ideal-supporting. Similarly, the algebra A has the CEP (congruence extension property) if for every subalgebra B the restriction map from congruences of A to those of B is surjective, in other words every congruence th of B can be extended to a congruence ph of A so that b th c iff b ph c for all b and c in B. This is also a little stronger property than ideal-supporting.

Looking up Hamiltonian and congruence on a web search leads to a 1991 paper of Ralph McKenzie (Algebra Universalis 28, Congruence Extension, Hamiltonian and Abelian properties in locally finite varieties) on the subject. It may not be the best starting place on a quest for ideal supporting varieties, but you may find it helpful.

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I love your epithets. – Taliberius 4 Feb 13 '13 at 22:04

It seems that the standard term for universal algebras whose congruences behave "as good as ideals" is "ideal-determined algebras". This is a more general notion than $\Omega$-group. This notion, along with its numerous particular cases and variations, was studied by Agliano, Chajda, Fichtner, Grätzer, Gumm, Slominski, Ursini and others. See, for example, I. Chajda, G. Eigenthaler, and H. Länger, Congruence Classes in Universal Algebra, Heldermann Verlag, 2003, Chapter 10.

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I suspect the term is simply to encompass "those algebras which support ideal constructions". Boolean algebras, for instance, have ideals that correspond exactly to the ring-theoretic notion.

There are generalisations of the ideal construction for wide classes of algebras. One form of generalised ideal uses the ability to define partial orders on particular algebras, often also using the fact that filters are dual to ideals. See, for instance, Shang and Li's generalisation to pseudo-effect algebras as a representative example, though that work also points to the definition on orthoalgebras and related.

There is also a not-so-common definition of ideal for algebras that

• the difference of elements in an algebra ideal are in the ideal
• alegbra multiplication of an element in the ideal and any element of the algebra is in the ideal
• scalar multiplication by an element of the ring on an element of the ideal is in the ideal

that is somewhat useful on it's own, though is more of a specialisation than a generalisation of the original ideal construction.

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