An axiomatic survey of diagram lemmas for non-abelian group-like structures

作者：Janelidze Zurab^*

来源：Journal of Algebra, 2012, 370: 387-401.

DOI：10.1016/j.jalgebra.2012.07.029

• Exact sequence
• Snake lemma
• 3 x 3 lemma
• Diagram chasing
• Ideal determined variety
• Normal category
• Subtractive category
• Subtraction

It is well known that diagram lemmas for abelian groups (and more generally in abelian categories) used in algebraic topology, can be suitably extended to %26quot;non-abelian%26quot; structures such as groups, rings, loops, etc. Moreover, they are equivalent to properties which arise in the axiomatic study of these structures. For the five lemma this is well known, and in the present paper we establish this for the snake lemma and the 3 x 3 lemma, which, when suitably formulated, turn out to be equivalent to each other for all (pointed) algebraic structures, and also in general categories of a special type. In particular, we show that among varieties of universal algebras, they are satisfied precisely in the so-called (pointed) ideal determined varieties.

• 出版日期2012-11-15