Lots of large graphs can be constructed from existing smaller graphs by using graph operations, such as the graph products. Many properties of such large graphs are closely related to those of the corresponding smaller ones.In this paper we consider some operations of Cayley graphs on semigroups.
A k-uniform hypergraph M is set-homogeneous if it is countable (possibly finite) and whenever two finite induced subhypergraphs U, V are isomorphic there is g ∈ Aut(M ) with U g = V ; the hypergraph M is said to be homogeneous if in addition every isomorphism between finite induced subhypergraphs extends to an automorphism. We give four examples of countably infinite set-homogeneous k-uniform hypergraphs which are not homogeneous (two with k = 3, one with k = 4, and one with k = 6). Evidence is also given that these may be the only ones, up to complementation. For example, for k = 3 there is just one countably infinite k-uniform hypergraph whose automorphism group is not 2-transitive, and there is none for k = 4. We also give an example of a finite set-homogeneous 3-uniform hypergraph which is not homogeneous.
A ‐uniform hypergraph is set‐homogeneous if it is countable (possibly finite) and whenever two finite induced subhypergraphs are isomorphic there is with ; the hypergraph is said to be homogeneous if in addition every isomorphism between finite induced subhypergraphs extends to an automorphism. We give four examples of countably infinite set‐homogeneous ‐uniform hypergraphs that are not homogeneous (two with , one with and one with ). Evidence is also given that these may be the only ones, up to complementation. For example, for there is just one countably infinite ‐uniform hypergraph whose automorphism group is not 2‐transitive, and there is none for . We also give an example of a finite set‐homogeneous 3‐uniform hypergraph that is not homogeneous.
By studying the semigroup presented by π = A, B | A n+1 = A, B m+1 = B, BA = A n−1 B, B m = A n , for every positive integers m, n ≥ 3 we show that, for even values of m this is an appropriate presentation for the semidirect product of monogenic semigroup S = a | a n+1 = a by the monogenic semigroup T = b | b m+1 = b. Moreover, for odd values of m it is a commutative semigroup of order (3+(−1) n)m 2. The interests of presentability of products of semigroups are quite useful in the study of more effective and intrinsic properties of the products (like as the computation of their characters and studying their Green J-classes). Our method of proof based on using the theoretic definitions and hereby we will deduced the presentation of the direct product of monogenic semigroups for every integers n, m ≥ 3.
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