Operads of hypergraphs.(English. Russian original)Zbl 1282.18008

Russ. Math. 57, No. 4, 52-62 (2013); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2013, No. 4, 61-73 (2013).
For all $$n \geq 1$$, and any monoid $$G$$, the authors construct two operad structures on the set of cubic $$n$$-multidimensional matrices. They both contain a suboperad $$HG_n$$ obtained from a symmetry condition on cubic multidimensional matrices. If $$G=\{0,1\}$$, with the monoid structure defined by $$1+1=1$$, these symmetric multidimensional matrices are the incidence matrices of hypergraphs such that any edge is adjacent to at most $$n$$ vertices, and the authors obtain in this way an operad structure on hypergraphs. It is also shown that these operads are $$Epi$$-operads, that is to say the right action of permutations can be extended to an action of surjections.

MSC:

 18D50 Operads (MSC2010) 05C25 Graphs and abstract algebra (groups, rings, fields, etc.)