Pair Of Homomorphisms Research Articles

Ordering coproducts of groups and semigroups

A. A. Vinogradov proved that a coproduct of orderable groups is orderable, and R. E. Johnson extended this result to semigroups. A simpler version of their proof is given, and results are obtained on orderability of group coproducts with amalgamation. In particular, it is shown that the coproduct of two or more copies of an orderable group H with amalgamation of a subgroup T⊆ H is orderable if and only if T is the difference kernel of a pair of homomorphisms from H into another orderable group, equivalently, if and only if T is convex under some one-sided group-ordering of H. For coproducts of families of nonisomorphic overgroups of a group T, partial results are obtained. Some related results are proved for one-sided orderability.

A representation of recursively enumerable languages by two homomorphisms and a quotient

A new representation for recursively enumerable languages is presented. It uses a pair of homomorphisms and the left (or right) quotient: For each recursively enumerable language L one can find homomorphisms h1, h2: ∑∗A → ∑∗B, such that w ∈ ∑∗L is a word in L if and only if w =h1(α)h2(α) for some α∈∑+A. (Or, each recursively enumerable language can be given by L = O(h1h2) ∩ ∑∗L, where O(h1h2) is the so-called right overflow languaged defined as O(h1h2) = {h1(x)h2(x); x ∈ ∑∗A}.)

Checking Sets, Test Sets, Rich Languages and Commutatively Closed Languages

The problem of homomorphism equivalence is to decide for some language L over some finite alphabet Σ and two homomorphisms f and g whether or not f ( x) = g( x) for all x in L. It has been conjectured that each L can be represented by some finite subset F such that for all pairs of homomorphisms f and g: f ( x) = g( x) for all x in F implies f ( x) = g( x) for all x in L. This conjecture is proved for the families of rich and commutatively closed languages. Lower and upper bounds are derived for the sizes of these finite subsets and examples of language families are given for which there are effective constructions of these subsets.

Equality Sets and Complexity Classes

If $h_1 $, $h_2 $ are two homomorphisms, then the equality set$\operatorname{Eq}(h_1 ,h_2 )$ of $h_1 $, $h_2 $ is $\operatorname{Eq} (h_1 ,h_2 ) = \{ w | h_1 (w) = h_2 (w)\} $. In this paper it is shown how to characterize complexity classes of formal languages in terms of equality sets of pairs of homomorphisms with bounded balance. In addition the complete twin shuffle language is investigated, and it is shown that for alphabets with at least two letters, this language cannot be represented as the equality set of a pair of homomorphisms unless both homomorphisms are erasing and have linear bounded balance.

Test sets for homomorphism equivalence on context free languages

We show that for every context free language L over some alphabet \gE there effectively exists a test set F , that is a finite subset of L such that, for any pair ( g, h ) of homomorphisms on \gE*, g ( x ) = h ( x ) for each x in F implies g ( x ) = h ( x ) for all x in L . This result is then extended from homomorphisms to deterministic generalized sequential machine mappings defined by machnies with uniformly bounded number of states.

Automorphisms of Trivalent Graphs

In [7] and [8], Tutte considered a vertex-transitive group of automorphisms of a finite, connected, trivalent graph. He showed that if the stabilizer of a is on adjacent vertices, then its order divides 3 . 24. As observed by Sims [5], the hypothesis is equivalent to the following group-theoretic conditions: a) G is a finite group generated by a pair of subgroups {P1, Pd, b) i Pi: Pi n P2 1= 3 for i = 1, 2, c) no non-trivial normal subgroup of G is contained in P1 A, d) P1 and P2 are G-conjugate. What happens if we drop condition d) or, what is essentially the same thing, replace vertex transitive by edge transitive? This question is primarily motivated by the examples afforded by the rank 2 BN pairs over GF(2). In this case, the trivalent graph mentioned above is the so-called building associated to the BN pair [6]. In this paper, we classify all pairs of subgroups (P1, P2) for which hypotheses a), b) and c) are satisfied. There are precisely fifteen such pairs, and in particular, we find that P1 n P2 has order dividing 27. In order to describe the results more completely, let us define an amalgam to be a pair of group monomorphisms (5, 02) with the same domain: P, 1 B P2. We will say that (01, 02) is finite if both co-domains P1, P2 are finite. In this case, we define the index of the amalgam to be the pair of indices (I Pl: im s1 I, P2: im 02 1). By a completion of the amalgam we mean a pair of homomorphisms (*1, '2) to some group G making the obvious diagram commute, i.e., such that 01* = 102'2 (in right-hand notation). By abuse of notation, we may say that G is a completion of the amalgam. Of course, we always have the trivial completion g1 = g2 = 1We also always have the universal completion, usually known as the amalgamated product, from

On epics and dominions of bands

If AFB is an extension of semigroups, then the dominion of A in B, Dom(A,B), is the set of all beB such that for each semigroup C and for each pair of homomorphisms f,g: B → C with f|A = g|A, then f(b) = g(b). It will be shown that epimorphisms are onto homomorphisms in the category of all bands with maximum conditions onD-classes. This contradicts an earlier report of a finite band B with a proper subband A such that Dom(A,B) = B. An example will be presented of a finite band B and a subband A such that Ac+Dom(A, B).

Zwei Klassen lokalkompakter maximal fastperiodischer Gruppen

In this paper we study the class\(\mathfrak{A}\) of all locally compact groupsG with the property that for each closed subgroupH ofG there exists a pair of homomorphisms into a compact group withH as coincidence set, and the class\(\mathfrak{D}\) of all locally compact groupG with the property that finite dimensional unitary representations of subgroups ofG can be extended to finite dimensional representations ofG. It is shown that [MOORE]-groups (every irreducible unitary representation is finite dimensional) have these two properties. A solvable group in\(\mathfrak{D}\) is a [MOORE]-group. Moreover, we prove a structure theorem for Lie groups in the class [MOORE], and show that compactly generated Lie groups in [MOORE] have faithful finite dimensional unitary representations.

T X is absolutely closed

If S, T are semigroups with S⊂T, then the dominion of S in T, Dom(S,T), is the set of all x e T such that for each semigroup U and for each pair of homomorphisms f,g: T→U with f|S=g|S, then f(x)=g(x). S is absolutely closed if Dom(S,T)=S for all T. That full transformation semigroups are absolutely closed has previously been reported. The intent here is to offer a corrected proof of that theorem.