A lattice ordered group(‘l-group’) is called complete if each set of elements that is bounded above has a least upper bound (and dually). A complete l-group is archimedean and hence abelian, and each archimedean l-group has a completion in the sense of the following theorem.

Metric regularity theory lies at the very heart of variational analysis, a relatively new discipline whose appearance was, to a large extent, determined by the needs of modern optimization theory in which such phenomena as nondifferentiability and set-valued mappings naturally appear. The roots of the theory go back to such fundamental results of the classical analysis as the implicit function theorem, Sard theorem and some others. This paper offers a survey of the state of the art of some principal parts of the theory along with a variety of its applications in analysis and optimization.

The paper considers the queueing system GI/G/1 with a type of customer impatience, namely, that the total queueing-time is uniformly limited. Using Lindiley's approach [10], an integral equation for the limiting waiting- time distribution is derived, and this is solved explicitly for M/G/1 using an expansion of the Pollaczek-Khintchine formula. It is also solved, in principle for Ej/G/l, and explicitly for Ej/Ek/l. A duality noted between GIA(x)/GB(x)/l and GIB(x)/GA(x)/l relates solutions for GI/Ek/l to Ek/G/l. Finally the equation for the busy period in GI/G/l is derived and related to the no-customer-loss distribution and dual distributions.

We give an explicit one step description of the free (Barr) exact category on a left exact one. As an application we give a new two step construction of the free abelian category on an additive one.

Let G be a group. The Fitting subgroup F(G) of G is defined to be the set union of all normal nilpotent subgroups of G. Since the product of two normal nilpotent subgroups is again a normal nilpotent subgroup (see [10] p. 238), F(G) is the unique maximal normal, locally nilpotent sungroup of G. In particular, is G is finite, then F(G) is the unique maximal normal nilpotent subgroup of G. If G is a notrivial solvable group, then clearly F(G) ≠1.

In this note we determine which of the trees homeomorphic to a star have a spectrum consisting entirely of integers. We also specify the integral double stars, and we consider the problem of trees with more complicated structure.

Subject classification (Amer. Math. Soc. (MOS) 1970): 05 C 05.

I establish here a criterion for a sequence of ordinals to be generic over a transitive model of ZFC with respect to a notation of forcing first considered by Prikry in his Doctoral dissertation [2]. In Section 0 I review some notation, in Section 1 I list some facts about measurable cardinals, and in Section 2, after giving Prikry's result, I state and prove mine.

An automorphism group of a graph is said to be s-regular if it acts regularly on the set of s-arcs in the graph. A graph is s-regular if its full automorphism group is s-regular. For a connected cubic symmetric graph X of order 2pn for an odd prime p, we show that if p ≠ 5, 7 then every Sylow p-subgroup of the full automorphism group Aut(X) of X is normal, and if p ≠3 then every s-regular subgroup of Aut(X) having a normal Sylow p-subgroup contains an (s − 1)-regular subgroup for each 1 ≦ s ≦ 5. As an application, we show that every connected cubic symmetric graph of order 2pn is a Cayley graph if p > 5 and we classify the s-regular cubic graphs of order 2p2 for each 1≦ s≦ 5 and each prime p. as a continuation of the authors' classification of 1-regular cubic graphs of order 2p2. The same classification of those of order 2p is also done.

This note gives a way of looking at Malcev conditions for varieties as ideals in a certain lattice. Though this viewpoint (so far) yields no new results, we feel it puts Walter Taylor's results [3] characterizing “Malcev definable classes of varieties” into a clearer perspective and is therefore worth mentioning. The author is grateful to R. Wille for pointing out W. Taylor's paper to him.

The Petersen graph on 10 vertices is the smallest example of a vertex-transitive graph which is not a Cayley graph. We consider the problem of determining the orders of such graphs. In this, the first of a series of papers, we present a sequence of constructions which solve the problem for many orders. In particular, such graphs exist for all orders divisible by a fourth power, and all even orders which are divisible by a square.

In [1] the concept of completeness of a functor was introduced and, in the cse of additive * categories and and an additive functor T: → , a criterion for T (supposed surjective) to be complete was given in terms of the kernel of T: this was that for each object A of the ideal A should be containded in the (Jacobson) radical of A. (The meaning of this notation and nomemclature is recalled in § 2 below). The question arises whether in any additive category there is a greatest ideal with this property, so that the canonical functor T: → / is in some sense the coarsest that faithfully represents the objects (but not the maps) of .

Consider a Galton-Watson process in which each individual reproduces independently of all others and has probability aj (j = 0, l, …) of giving rise to j progeny in the following generation and in which there is an independent immigration component where bj (j = 0, l, …) is the probability that j individuals enter the population at each generation. Then letting Xn (n = 0, l, …) be the population size of the n-th generation, it is known (Heathcote [4], [51]) that {Xn} defines a Markov chain on the non-negative integers. Unless otherwise stated, we shall consider only those offspring and immigration distributions that make the Markov chain {Xn} irreducible and aperiodic.

An approximate dual is proposed for a multiobjective optimization problem. The approximate dual has a finite feasible set, and is constructed without using a perturbation. An approximate weak duality theorem and an approximate strong duality theorem are obtained, and also an approximate variational inequality condition for efficient multiobjective solutions.

A partially ordered semigroup S is said to be a Dubreil-Jacotin semigroup if there is an isotone homomorphism θ of S onto a partially ordered group such that {} has a greatest member. In this paper we investigate the structure of regular Dubreil-Jacotin semigroups in which the imposed partial order extends the natural partial order on the idempotents. The main tool used is a local structure theorem which is introduced in Section 2. This local structure theorem applies to many other contexts as well.

This is a study of formal power series under the binary operation of formal composition from a group-theoretical point of view. Various “large” properties are derived.

Let Λ be the set of inequivalent representations of a finite group over a field . Λ is made the basis of an algebra over the complex numbers , called the representation algebra, in which multiplication corresponds to the tensor product of representations and addition to direct sum. Green [5] has shown that if char (the non-modular case) or if is cyclic, then is semi-simple, i.e. is a direct sum of copies of . Here we consider two modular, non-cyclic cases, viz, where is or 4 (alternating group) and is of characteristic 2.

Hadamard matrices exist for infinitely many orders 4m, m ≧ 1, m integer, including all 4m < 100, cf. [3], [2]. They are conjectured to exist for all such orders. Skew Hadamard matrices have been constructed for all orders 4m < 100 except for 36, 52, 76, 92, cf. the table in [4]. Recently Szekeres [6] found skew Hadamard matrices of the order 2(pt +1)≡ 12 (mod 16), p prime, thus covering the case 76. In addition, Blatt and Szekeres [1] constructed one of order 52. The present note contains a skew Hadamard matrix of order 36 (and one of order 52), thus leaving 92 as the smallest open case.

A completely regular semigroup is a semigroup which is a union of groups. The class CR of completely regular semigroups forms a variety. On the lattice L (CR) of completely regular semigroup varieties we define two closure operations which induce complete congruences. The consideration of a third complete congruence on L (CR) yields a subdirect decomposition of L (CR). Using these results we show that L (CR) is arguesian. This confirms the (tacit) conjecture that L (CR) is modular.

We show that every σ-complete MV-algebra is an MV-σ-homomorphic image of some σ-complete MV- algebra of fuzzy sets, called a tribe, which is a system of fuzzy sets of a crisp set Ω containing 1Ω and closed under fuzzy complementation and formation of min {∑nfn, 1}. Since a tribe is a direct generalization of a σ-algebra of crisp subsets, the representation theorem is an analogue of the Loomis-Sikorski theorem for MV-algebras. In addition, this result will be extended also for Dedekind σ-complete ℓ-groups with strong unit.

The aim of the paper is to prove an elliptic analogue of a deep theorem of Iwasawa on cyclotomic fields.

Subject classification(Amer. Math. Soc. (MOS) 1970): primary 12 A 35, 12 A 65.