Finite Paths are Universal

We prove that any countable (finite or infinite) partially ordered set may be represented by finite oriented paths ordered by the existence of homomorphism between them. This (what we believe a surprising result) solves several open problems. Such path-representations were previously known only for finite and infinite partial orders of dimension 2. Path-representation implies the universality of other classes of graphs (such as connected cubic planar graphs). It also implies that finite partially ordered sets are on-line representable by paths and their homomorphisms. This leads to new on-line dimensions. Mathematics Subject Classifications (2000) 06A06, 06A07, 05E99, 05C99.J. Nešetřil: Supported by a Grant LN00A56 of the Czech Ministry of Education. The first author was partially supported by EU network COMBSTRU at UPC Barcelona.     

Quotients of Some Finite Universal Locally Projective Polytopes

   Abstract. This paper classifies the quotients of a finite and locally projective polytope of type {4,3,5} . Seventy quotients are found, including three regular polytopes, and nine other section regular polytopes which are themselves locally projective. The classification is done with the assistance of GAP, a computer system for algebraic computation. The same techniques are also applied to two finite locally projective polytopes respectively of type {3,5,3} and {5,3,5} . No nontrivial quotients of the latter are found.     

Quotients of Some Finite Universal Locally Projective Polytopes

Abstract. This paper classifies the quotients of a finite and locally projective polytope of type {4,3,5} . Seventy quotients are found, including three regular polytopes, and nine other section regular polytopes which are themselves locally projective. The classification is done with the assistance of GAP, a computer system for algebraic computation. The same techniques are also applied to two finite locally projective polytopes respectively of type {3,5,3} and {5,3,5} . No nontrivial quotients of the latter are found.     

分支问题的有限决定性和万有开折

本文给出了分支问题有限决定的一些充分必要条件，并用横切性条件刻划了分支问题的万有开折，证明了分支问题的万有开折与无穷小稳定是等价的     

Finite CI-Groups are Soluble

For a finite group G and a subset S of G with 1 S and S = S^–1,the Cayley graph Cay(G, S) is the graph with vertex set G suchthat {x, y} is an edge if and only if yx^–1 S. The groupG is called a CI-group if, for all subsets S and T of G\{1},Cay(G, S) Cay(G, T) if and only if S = T for some Aut(G).In this paper, for each prime p 1 (mod 4), a symmetric graph(p) is constructed from PSL(2, p) such that Aut (p) = Z₂ x PSL(2,p); it is then shown that A₅ is not a CI-group, and that allfinite CI-groups are soluble. 1991 Mathematics Subject Classification05C25.     

Minimal Paths between Maximal Chains in Finite Rank Semimodular Lattices

We study paths between maximal chains, or flags, in finite rank semimodular lattices. Two flags are adjacent if they differ on at most one rank. A path is a sequence of flags in which consecutive flags are adjacent. We study the union of all flags on at least one minimum length path connecting two flags in the lattice. This is a subposet of the original lattice. If the lattice is modular, the subposet is equal to the sublattice generated by the flags. It is a distributive lattice which is determined by the Jordan-Hölder permutation between the flags. The minimal paths correspond to all reduced decompositions of this permutation. In a semimodular lattice, the subposet is not uniquely determined by the Jordan-Hölder permutation for the flags. However, it is a join sublattice of the distributive lattice corresponding to this permutation. It is semimodular, unlike the lattice generated by the two flags, which may not be ranked. The minimal paths correspond to some reduced decompositions of the permutation, though not necessarily all. We classify the possible lattices which can arise in this way, and characterize all possibilities for the set of shortest paths between two flags in a semimodular lattice.     

Finite bands are finitely related

We prove that every finite idempotent semigroup (band) is finitely related, which means that the clone of its term operations (i.e. operations induced by words) is determined by finitely many relations. This solves an open problem posed by Mayr (Semigroup Forum 86:613–633, 2013).     

Universal kernels which are continuous on the diagonal

Positivity - Let G be a locally compact abelian group and X be a nonempty compact set of G. Given a positive definite kernel $$K:G times G rightarrow {mathbb {C}}$$ whose real part is continuous...     

Finite nilpotent rings are not dualizable

In this paper we show that a finite nilpotent ring that is not a zero-ring cannot admit a natural duality. In fact, every finite ring having a nilpotent subring (which is nilpotent of class ≥ 2) is not dualizable. Received October 28, 1998; accepted in final form July 8, 1999.     

Finite fixed point algebras are subdiagonalisable

Presented by Joel Berman.     

Finite groups are big as semigroups

We prove that a finite group G occurs as a maximal proper subsemigroup of an infinite semigroup (in the terminology of Freese, Ježek, and Nation, G is a big semigroup) if and only if |G| ≥ 3. In fact, any finite semigroup whose minimal ideal contains a subgroup with at least three elements is big.     

Frattini Theory for Classes of Finite Universal Algebras of Malcev Varieties

We extend the Frattini theory of formations and Schunck classes of finite groups to some Frattini theory of formations and Schunck classes of finite universal algebras of Malcev varieties. We prove that if F(1) is a nonempty formation (Schunck class) of algebras of a Malcev variety, then its Frattini subformation (Frattini Schunck subclass) (F) consists of all nongenerators of F; moreover, if M is a formation (Schunck class) in F; then (M) (F).     

Groups Whose Finite Homomorphic Images are Metahamiltonian

A group G is metahamiltonian if all its non-abelian subgroups are normal. It is proved here that a finitely generated soluble group is metahamiltonian if and only if all its finite homomorphic images are metahamiltonian; the behaviour of soluble minimax groups with metahamiltonian finite homomorphic images is also investigated. Moreover, groups satisfying the minimal condition on non-metahamiltonian subgroups are described.     

Finite Groups Whose Minimal Subgroups are c-Supplemented

Let G be a finite group. A subgroup H of a group G is said to be c-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ≤ H _G , where H _G  = Core _G (H) is the largest normal subgroup of G contained in H. In this article, we investigate the structure of a finite group G under the assumption that subgroups of prime order are c-supplemented in G. Moreover, we analyze the structure of a group G when the minimal subgroups of the generalized Fitting subgroup F?(G) of G are c-supplemented in G through the theory of formations.     

Finite p-groups whose nonnormal subgroups are metacyclic

For an odd prime p, we give a criterion for finite p-groups whose nonnormal subgroups are metacyclic, and based on the criterion, the p-groups whose nonnormal subgroups are metacyclic are classified up to isomorphism. This solves a problem proposed by Berkovich.