Completely dissociative groupoids

Abstract

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Associative spectra of graph algebras I

“…It was also proved in [14] that there exist a continuum of different associative spectra (allowing infinite base sets, of course). Similar questions were investigated in [2][3][4], where some of the earlier results were rediscovered (with a different terminology).…”

“…We have proved so far that s n (A(C m )) equals the number of zag sequences that satisfy (2). Note the difference between (1) and ( 2): an arbitrary zag sequence can have arbitrarily large decreases, while a sequence satisfying (2) can drop at most by m − 2.…”

“…. , n 2. It is straightforward to verify that β(d) is also a zag sequence, and every zag sequence bounded above by h + 1 is in the image ofβ (indeed, if d is bounded by h + 1, then β(d) = d).…”

Associative spectra of graph algebras I. Foundations, undirected graphs, antiassociative graphs

“…Graph algebras were introduced by C. R. Shallon [17]. We associate any digraph G = (V, E) with an algebra A(G) = (V ∪ {∞}; •, ∞) of type (2,0), where ∞ is a new element distinct from the vertices, and the binary operation is defined by the following rule: for any x, y ∈ V ∪ {∞},…”

“…Proof. Homomorphisms of a DFS tree T into G are uniquely determined by the image of x 1 : a map ϕ : X n → V (G) with ϕ(x 1 ) = v k a homomorphism from 2 The map β can be explained in terms of DFS trees as follows: if T is the DFS tree corresponding to the zag sequence d, then β(d) corresponds to the DFS tree obtained from T by turning, for each vertex v at depth h, all descendants of v into children of v.…”