Template talk:Group-like structures

Closure Property

Hello!

Why doesn't categories and groupoids have the closure property? I think, they are rather missing the totality property.

—Markus Prokott (talk) 07:10, 19 April 2008 (UTC)

I think because not every pair of arrows are composabe (i.e. $f:A\to B,g:C\to D$) --Dima Gerasimov (talk) 09:53, 6 November 2013 (UTC)

Template overhaul: remaining uncertainty

I really think that the last three entries should be somehow separated from the main table with the note that they pertain more to category theory, as opposed to fundamental group theory. Since I am less than familiar with the basics of category theory, perhaps someone more knowledgable on this could review and revise (if necessary)? DWIII (talk) 20:30, 28 April 2012 (UTC)

I want to note that their inclusion in this table can be misleading since an arbitrary Monoid without totality doesn't automatically give rise to a Category, at least not using the classical definition. 87.165.42.165 (talk) 13:15, 2 July 2012 (UTC)

Minor Edits

I have corrected some entries to point out that elements of quasigroups and loops do not have inverses; but in a loop, every element has a left inverse and a right inverse. The older 'talk' entries reveal that someone had altered these entries in an attempt to force the quasigroup row entries of this table to differ from the row entries for general magmas. Actually the difference between a quasigroup and a general magma (the fact that in a quasigroup, each left multiplication map and right multiplication map is bijective) is not relevant to this table.

There are many more classes of magmas, and possible properties of magmas, that might be considered for additional inclusion in this table; and it does not seem reasonable to add too many more rows or columns. However given the current popularity of hypergroups, I wonder about trying to include them in this list of group-like structures. Gemisi (talk) 20:56, 19 August 2013 (UTC)

Loops are quasigroups with identity element, shouldn't they have Yes** in the "Inverses" column like loops? --Dima Gerasimov (talk) 09:42, 6 November 2013 (UTC)

"Semicategory"?

Is 'semicategory' the standard term for a set with a partial associative binary operation? As the name for a set endowed with a total associative binary operation is called a 'semigroup', and when we remove the axiom of identity from the definition of a monoid, we do not call it a 'semimonoid'. Thus, calling its non-closed cousin a 'semicategory' is inconsistent nomenclature. I propose we use the term 'semigroupoid' here and mention any alternate names on the target page. Further supporting this suggestion is, a) the 'semicategory' link actually redirects to the 'semigroupoid' page anyway; b) on a lesser note, 'semigroupoid' is the preferred term for the Haskell implementation of this algebraic structure. MaxwellEdisonPhD (talk) 22:35, 3 October 2013 (UTC)

It might not be a non-standard term for a set with a partial associative binary operation, but this term does exist in the cathegory theory. However, this indeed confuses and we could split the table in two according to this proposal. --Dima Gerasimov (talk) 10:07, 6 November 2013 (UTC)

Cancellation instead of Invertibility?

The Magma (algebra) page constructs a quasigroup from a magma by adding the divisibility axiom, and shows this in a convienient summary diagram. As either the divisibility axiom or the invertibility axiom imply the cancellation property, should we consider having a Cancellation column instead of an Invertibility column? This would neatly emphasise the pattern centered on the abelian group row of the table, which aids understanding of the specification of group-like objects by their axioms. As a suggestion, we could do it like this:

	Cancellation
Magma	No
Semigroup	No
Monoid	No
Group	by Inverses
Abelian Group	by Inverses
Loop	by Divisibility²
Quasigroup	by Inverses
Groupoid	No
Category	No
Semigroupoid	No

² each element of a loop has a left- and right-inverse, but these need not co-incide.

MaxwellEdisonPhD (talk) 23:10, 3 October 2013 (UTC)

The point is well-made, but cancellability is not strong enough. One can have cancellability without division being defined as a total function. I have changed this to 'Division'. 172.82.47.18 (talk) 12:41, 28 April 2022 (UTC)

Closure vs Totality

The explanation that "Closure, which is used in many sources, is an equivalent axiom to totality, though defined differently" leaves a lot to be desired. Especially since the link for Totality just goes to a small paragraph on Total Functions on the Partial Functions page, which says Total Functions are a synonym for "functions". That to me says nothing about the closure of an operation over a set. (Which in terms of functions means something like the range is the same as the domain.) Wmhilton (talk) 17:15, 29 January 2016 (UTC)

'Divisibility' changed to 'invertibility'

I am about to change the heading of the 'Divisibility' column to 'Invertibility'. That is because, self-evidently, someone reading a table of 'Group-like structures' is more likely to be familiar with groups than any of the other structures, and the group axioms are usually described as closure, associativity, identity and invertibility, including in the lede to Group (mathematics). Readers who are learning about groups for the first time are likely to be confused by the discrepancy between this column heading and the lede (see eg. Talk:Group (mathematics)#Division.3F), and also by the fact that clicking on the heading leads to an article about 'inverses' rather than 'division.'

I appreciate that invertibility entails divisibility in the sense that the group binary operation, which may be arbitrarily called 'multiplication', is invertible. But again, it is confusing to refer to this as 'divisibility' when the integers under addition are commonly identified as the canonical example of a group, and 'multiplication' and 'division' are specifically used in this context to refer to a binary operation which has only a partial inverse and cannot therefore satisfy the group axioms, by analogy with integer multiplication (see eg. division ring).

If you disagree and would like to revert my change, I would appreciate it if you could let me know your reasons for doing so. Thanks. splintax ^(talk) 10:27, 28 September 2016 (UTC)

I have changed this to 'Division'. This column really is about the structure having the axiom that defines a quasigroup: the latin square property. This is a stronger property than cancellability, but I do not know of a better way of compactly naming the property. Division rings are named by this property on a proper subset (the set with 0 excluded). It is definitely not invertibility, which implies an identity element, which is not true of quasigroups in general. 172.82.47.18 (talk) 12:34, 28 April 2022 (UTC)

Removed 'Inverse Semigroup'

I removed the Inverse semigroup row because it seems to be misleading. An inverse semigroup with a unit is an inverse monoid, not a group; an associative quasigroup is a (possibly empty) group, not an inverse semigroup. Also, it's not clear that the pseudo-inverse property that inverse semigroups have is the most analogous; perhaps "regular semigroup" might be better, if it's also noted that the invertibility criterion here is different. Elmach2MN2 (talk) 07:26, 28 September 2021 (UTC)

Your statements are in contradiction with this picture: https://upload.wikimedia.org/wikipedia/commons/thumb/0/07/Algebraic_structures_-_magma_to_group.svg/800px-Algebraic_structures_-_magma_to_group.svg.png 213.22.81.218 (talk) 17:43, 11 November 2022 (UTC)

See also here: https://commons.wikimedia.org/wiki/File_talk:Magma_to_group4.svg 213.22.81.218 (talk) 17:55, 11 November 2022 (UTC)