Geodesic
Combinatorial encodings of infinite symmetric groups and descriptions of semigroups of double cosets
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXVIII, Tome 462 (2017), pp. 65-92

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Spaces of double cosets of infinite symmetric groups with respect to some special subgroups admit natural structures of semigroups. Elements of such semigroups can be interpreted in combinatorial terms. We present a description of such constructions in a relatively wide degree of generality. 
@article{ZNSL_2017_462_a3,
     author = {Yu. A. Neretin},
     title = {Combinatorial encodings of infinite symmetric groups and descriptions of semigroups of double cosets},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {65--92},
     publisher = {mathdoc},
     volume = {462},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_462_a3/}
}
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Yu. A. Neretin. Combinatorial encodings of infinite symmetric groups and descriptions of semigroups of double cosets. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXVIII, Tome 462 (2017), pp. 65-92. http://geodesic.mathdoc.fr/item/ZNSL_2017_462_a3/