A New Characterisation of Groups Amongst Monoids

Abstract

Abstract. We prove that a monoid M is a group if and only if, in the category of monoids, all points over M are strong. This sharpens and greatly simplifies a result of Montoli, Rodelo and Van der Linden [8] which characterises groups amongst monoids as the protomodular objects.In their article [8], Montoli, Rodelo and Van der Linden introduce, amongst other things, the concept of a protomodular object in a finitely complete category C as an object Y P C over which all points are stably strong. The aim of their definition is two-fold: first of all, to provide a categorical-algebraic characterisation of groups amongst monoids as the protomodular objects in the category Mon of monoids; and secondly, to establish an object-wise approach to certain important conditions occurring in categorial algebra such as protomodularity [2, 1] and the Mal'tsev axiom [5,6].We briefly recall some basic definitions; see [3,7,8] for more details. Let C be a finitely complete category, which we also take to be pointed for the sake of simplicity. In C , a pair of arrows pr : W Ñ X, s : Y Ñ Xq is jointly strongly epimorphic when if mr 1 " r, ms 1 " s for some given monomorphism m : M Ñ X and arrows r 1 : W Ñ M , s 1 : Y Ñ M , then m is an isomorphism. In the case of monoids, this means that any x P X can be written as a product rpw 1 qspy 1 q¨¨¨rpw n qspy n q for some w j P W , y j P Y . This characterisation follows easily from the fact that pr, sq is a jointly strongly epimorphic pair in Mon if and only if the induced monoid morphism W`Y Ñ X is a surjection-see, for instance, [1, Corollary A.5.4 combined with Example A.5.16]. Given an object Y in C , a point over Y is a pair of morphisms pf : X Ñ Y , s : Y Ñ Xq such that f s " 1 Y . A point pf, sq is said to be strong when the pair pkerpf q : Kerpf q Ñ X, s : Y Ñ Xq is jointly strongly epimorphic. The point pf, sq is stably strong when all of its pullbacks are strong. More precisely, if g : Z Ñ Y is any morphism, then the pullback g˚pf q together with its splitting induced by s is a strong point.Even though the concept of a protomodular object serves the intended purpose of characterising groups amongst monoids, the proof of this characterisation given in [8] is rather complicated, since it relies on another, more subtle, characterisation in terms of the so-called Mal'tsev objects. The present short note aims to improve the situation by giving a quick and direct proof of a more general result: a monoid is a group as soon as all points over it are strong. Theorem. A monoid M is a group if and only if, in Mon, all points over M are strong.

A note on split extensions of bialgebras

“…All split extensions over a bialgebra Y admit a join decomposition if and only if Y is a Hopf algebra. This result is along the lines of, and is actually a variation on, a similar characterization of groups among monoids, recently obtained in [12,7]. There the authors show 2010 Mathematics Subject Classification.…”

“…Applying the functor G, we regain the original split extension, since G is a right adjoint, thus preserves kernels; but G also preserves jointly extremally epimorphic pairs by Proposition 3.3, so that the pair pk, sq is jointly extremally epimorphic. As a consequence, all split extensions over the monoid GpY q are strong, and GpY q is protomodular [7].…”

“…In the case of groups and monoids (C " Set in Theorem 2.4), it was shown in [12] (resp. in [7]) that all points in Mon over Y are stably strong (resp. strong) if and only if Y is a group.…”

“…Then Proposition 3.4 implies that in Mon all split extensions over GpY q are strong. Hence GpY q is a group by the result in [7], which makes Y a Hopf algebra by [14, 8.0.1.c and 9.2.5].…”

On stability of exactness properties under the pro-completion

“…A related line of research is to identify exactness properties of an object which describe certain algebraically defined members of a variety. A striking positive result in this area is given in [77,40], where it is shown that groups can be identified in the variety of monoids using a first-order exactness property.…”