We prove that the lattice EqΩ of all equivalence relations on an infinite set Ω contains, as a 0, 1-sublattice, the 0-coproduct of two copies of itself, thus answering a question by G.M. Bergman. Hence, by using methods initiated by de Bruijn and further developed by Bergman, we obtain that EqΩ also contains, as a sublattice, the coproduct of 2 Ω copies of itself.

Let M be a closed oriented d-dimensionnal manifold and let LM be the space of free loops on M . In this paper, we give a geometrical interpretation of the loop-coproduct and we study it’s compatibility with the Serre spectral sequence associated to the fibration ΩM → LM ev(0) → M . Then, we show that the spectral sequence associated to the free loop fibration LN → LX → LM of some Serre fibratio...

The production and use of biofuel have increased under the context of sustainable development. Biofuel production from plant biomass not only produces biofuel or ethanol but also coproduces products containing lignin, modified lignin, and lignin derivatives. The use of lignin-containing biofuel coproduct in pavement soil stabilization was explored as a new application area. The primary experime...

in 2001, s. bulman-fleming et al. initiated the study of three flatness properties (weakly kernel flat, principally weakly kernel flat, translation kernel flat) of right acts $a_{s}$ over a monoid $s$ that can be described by means of when the functor $a_{s} otimes -$ preserves pullbacks. in this paper, we extend these results to$s$-posets and present equivalent descriptions of weakly kernel po...

Let k be a commutative ring and let C be the operad of differential graded k-modules obtained as the singular k-chains of the linear isometries operad [4, §V.9]. We show that the category of C-algebras is a proper closed model category. We use the amenable description of the coproduct in this category [4, V.3.4] to analyze the coproduct of and develop a homotopy theory for algebras over an arbi...

q-Matroids form the q-analogue of classical matroids. In this paper we introduce various types maps between q-matroids. These are not necessarily linear, but they map subspaces to and respect q-matroid structure in certain ways. The give rise different categories We show that only one these possesses a coproduct. This is category where morphisms linear weak maps, is, rank image any subspace lar...

The homology of binary 3–dimensional digital images (digital volumes) provides concise algebraic description of their topology in terms of connected components, tunnels and cavities. Homology generators corresponding to these features are represented by nontrivial 0– cycles, 1–cycles and 2–cycles, respectively. In the framework of cubical representation of digital volumes with the topology that...

The coproduct is given by summing over self-intersections and removing the intersection point, and smoothing to obtain two paths. Precisely, let ∩α denote the set of self-intersections of α (assuming α is in general position). The coproduct is then given by summing over selfintersections and removing the intersection point, and smoothing to obtain two paths. Then δ(〈α〉) = ∑ q∈∩α 〈α1 q〉 ∧ 〈α2 q〉...