Idempotence

About: Idempotence is a research topic. Over the lifetime, 1860 publications have been published within this topic receiving 19976 citations. The topic is also known as: idempotent.

Idempotent Multiplications on Surfaces and Aspherical Spaces

Abstract: Every continuous idempotent multiplication on a space induces an idempotent comultiplication on its cohomology algebra over a commutative ring and a homomorphic idempotent multiplication on each homotopy group. We classify all idempotent comultiplications on any graded anticommutative algebra A∗ over a principal ideal domain K up to degree 2 provided the degree 1 component A1 is torsion free and the degree 2 component A2 is of rank 1. All algebraic possibilities can be topologically realized. We also describe all homomorphic idempotent multiplications on arbitrary groups. This allows a complete classification up to homotopy of all idempotent multiplications on aspherical CW-complexes. For surfaces we obtain an explicit list. Notably, the Klein bottle allows infinitely many nonhomotopic idempotent multiplications, but all other surfaces with nonabelian fundamental group have only the projections as idempotent multiplications (up to homotopy). Introduction. Idempotent multiplications on sets and topological spaces have been considered by many authors, for instance as an axiomatic approach to the averaging operation (sample: [2, 3, 10]). If X denotes a connected topological space, then the existence of H-space structures places severe restrictions on the structure of X. (See, for instance, [6] or [16].) This is due to the presence of homotopy identities on both sides. If, however, one considers idempotent multiplications μ : X ×X → X, that is, multiplications which satisfy μ(x, x) = x for all x ∈ X, then no restriction follows from the presence of such multiplications, since every space X allows the two idempotent multiplications p1, p2 : X × X → X, p(x, y) = x and q(x, y) = y for all x, y ∈ X. These are the so-called trivial multiplications. On the other hand, the existence of nontrivial idempotent multiplication again forces restrictions on the space. We wish to illustrate this by discussing idempotent multiplications on suitable classes of spaces. The Received by the editors on September 1, 1988, and in revised form on October 24, 1988. Copyright c ©1991 Rocky Mountain Mathematics Consortium

Bounds in the idempotent theorem

Abstract: We show that if G is a finite Abelian group and f is an integer-valued map on G with algebra norm at most M then there is some L < \exp(M^{4+o(1)}), cosets of (possibly different) subgroups W_1,,W_L, and s_1,,s_L \in {-1,1} such that f=\sum_i{s_i1_{W_i}}

On Rank Properties of Endomorphisms of Finite Circular Orders

Abstract: A mapping α from X = {1,2,…,n} to X is orientation-preserving if the sequence (1α,2α,…,nα) is a cyclic permutation of a nondecreasing sequence (with respect to some total order on X). Orientation-preserving mappings can be thought of as preserving a circular order on X. Two partitions of X have the same type if they have identical sizes and numbers of classes. Let τ be a partition with r classes, and let S be the semigroup generated by the set of orientation-preserving mappings from X to X with kernels of same type as τ. We show that the minimum cardinality of a generating set for S is . Moreover, we characterize all such S generated by their idempotent elements (i.e., s ∊ S such that s 2 = s), and show that the minimum number of idempotent elements required to generate S is .

Standard Bases of a Vector Space Over a Linearly Ordered Incline

Abstract: Inclines are additively idempotent semirings, in which the partial order ≤ : x ≤ y if and only if x + y = y is defined and products are less than or equal to either factor. Boolean algebra, max-min fuzzy algebra, and distributive lattices are examples of inclines. In this article, standard bases of a finitely generated vector space over a linearly ordered commutative incline are studied. We obtain that if a standard basis exists, then it is unique. In particular, if the incline is solvable or multiplicatively-declined or multiplicatively-idempotent (i.e., a chain semiring), further results are obtained, respectively. For a chain semiring a checkable condition for distinguishing if a basis is standard is given. Based on the condition an algorithm for computing the standard basis is described.