Search

In 2013, Farhi conjectured that for each $m\geq 3$, every natural number $n$ can be represented as $\lfloor x^2/m\rfloor+\lfloor y^2/m\rfloor+\lfloor z^2/m\rfloor$ with $x,y,z\in\Z$, where $\lfloor\cdot\rfloor$ denotes the floor function. Moreover, in 2015, Sun conjectured that every natural number $n$ can be written as $\lfloor x^2/a\rfloor+\lfloor y^2/b\rfloor+\lfloor z^2/c\rfloor$ with $x,y,z\in\Z$, where $a,b,c$ are integers and $(a,b,c)\neq (1,1,1),(2,2,2)$. In this paper, with the help of congruence theta functions, we prove that for each $m\geq 3$, Farhi's conjecture is true for every sufficiently large integer $n$. And for $a,b,c\geq 5$ with $a,b,c$ are pairwisely co-prime, we also confirm Sun's conjecture for every sufficiently large integer $n$.

We prove a number of unconditional statistical results of the Hecke coefficients for unitary cuspidal representations of GL ( 2 ) \text {GL}(2) over number fields. Using partial bounds on the size of the Hecke coefficients, instances of Langlands functoriality, and properties of Rankin–Selberg L L -functions, we obtain bounds on the set of places where linear combinations of Hecke coefficients are negative. Under a mild functoriality assumption we extend these methods to GL ( n ) \text {GL}(n) . As an application, we obtain a result related to a question of Serre about the occurrence of large Hecke eigenvalues of Maass forms. Furthermore, in the cases where the Ramanujan conjecture is satisfied, we obtain distributional results of the Hecke coefficients at places varying in certain congruence or Galois classes.

In this paper we discuss the existence of certain classes of cuspidal automorphic representations having non-zero Fourier coefficients for a general semisimple algebraic group $ G$ defined over a number field $ k$ such that its Archimedean group $ G_\infty $ is not compact. When $ G$ is quasi-split over $ k$, we obtain a result on existence of generic cuspidal automorphic representations which generalize results of Vignéras, Henniart, and Shahidi. We also discuss: (i) the existence of cuspidal automorphic forms with non-zero Fourier coefficients for congruence of subgroups of $ G_\infty $, and (ii) applications related to the work of Bushnell and Henniart on generalized Whittaker models.

In this article we prove a congruence formula for the special values of certain dihedral twists of two primitive modular forms of weight two with isomorphic residual Galois representation at a prime p p .

If G is a GGS-group defined over a p-adic tree, where p is an odd prime, we calculate the order of the congruence quotients $G_n=G/\Stab_G(n)$ for every n. If G is defined by the vector $e=(e_1,...,e_{p-1})\in\F_p^{p-1}$, the determination of the order of $G_n$ is split into three cases, according as e is non-symmetric, non-constant symmetric, or constant. The formulas that we obtain only depend on p, n, and the rank of the circulant matrix whose first row is e. As a consequence of these formulas, we also obtain the Hausdorff dimension of the closures of all GGS-groups over the p-adic tree., Comment: 27 pages

Let p be any prime, and let a and n be nonnegative integers. Let $r\in Z$ and $f(x)\in Z[x]$. We establish the congruence $$p^{\deg f}\sum_{k=r(mod p^a)}\binom{n}{k}(-1)^k f((k-r)/p^a) =0 (mod p^{\sum_{i=a}^{\infty}[n/p^i]})$$ (motivated by a conjecture arising from algebraic topology), and obtain the following vast generalization of Lucas' theorem: If a is greater than one, and $l,s,t$ are nonnegative integers with $s,t

Fernando Rodriguez-Villegas has conjectured a number of supercongruences for hypergeometric Calabi-Yau manifolds of dimension d less than or equal to 3. For manifolds of dimension d = 1, he observed four potential supercongruences. Later the author proved one of the four. Motivated by Rodriguez-Villegas's work, in the present paper we prove a general result on supercongruences between values of truncated F-2(1) hypergeometric functions and Gaussian hypergeometric functions. As a corollary to that result, we prove the three remaining supercongruences.

We give a Lorentzian metric on the null congruence associated with a timelike conformal vector field. A Liouville type theorem is proved and a boundedness for the volume of the null congruence, analogous to a well-known Berger-Green theorem in the Riemannian case, will be derived by studying conjugate points along null geodesics. As a consequence, several classification results on certain compact Lorentzian manifolds without conjugate points on its null geodesics are obtained. Finally, several properties of null geodesics of a natural Lorentzian metric on each odd-dimensional sphere have been found.

For every finitely generated, congruence modular varietyV\mathcal {V}of finite type we find a finite familyR\mathcal {R}of finite rings such that the varietyV\mathcal {V}is finitely decidable if and only ifV\mathcal {V}is congruence permutable and residually small, all solvable congruences in finite algebras fromV\mathcal {V}are Abelian, each congruence above the centralizer of the monolith of a subdirectly irreducible algebraA\mathbf {A}fromV\mathcal {V}is comparable with all congruences ofA\mathbf {A}, each homomorphic image of a subdirectly irreducible algebra with a non-Abelian monolith has a non-Abelian monolith, and, for each ringRRfromR\mathcal {R}, the variety ofRR–modules is finitely decidable.

The space of congruence classes of full spherical minimal immersions f : S m → S n f:S^m\to S^n of a given source dimension m m and algebraic degree p p is a compact convex body M m p \mathcal {M}_m^p in a representation space F m p \mathcal {F}_m^p of the special orthogonal group S O ( m + 1 ) SO(m+1) . In Ann. of Math. 93 (1971), 43–62 DoCarmo and Wallach gave a lower bound for F m p \mathcal {F}_m^p and conjectured that the estimate was sharp. Toth resolved this “exact dimension conjecture” positively so that all irreducible components of F m p \mathcal {F}_m^p became known. The purpose of the present paper is to characterize each irreducible component V V of F m p \mathcal {F}_m^p in terms of the spherical minimal immersions represented by the slice V ∩ M m p V\cap \mathcal {M}_m^p . Using this geometric insight, the recent examples of DeTurck and Ziller are located within M m p \mathcal {M}_m^p .

We discuss the structure of the ideal class group of real biquadratic fields K K , concentrating on the case that the 4 4 -rank of the ideal class groups of the quadratic subfields of K K is 0 0 . In this case, we give estimates for the 4 4 -rank of the ideal class group of K K . As an example, let K = Q ( p , 627 ) K = \mathbb {Q}(\sqrt p ,\sqrt {627} ) , where p p is a prime satisfying certain congruence conditions. The 2 2 -primary part of the ideal class group of K K is then isomorphic to ( Z / 4 Z ) 2 , Z / 4 Z × ( Z / 2 Z ) 2 {(\mathbb {Z}/4\mathbb {Z})^2},\mathbb {Z}/4\mathbb {Z} \times {(\mathbb {Z}/2\mathbb {Z})^2} , or ( Z / 2 Z ) 4 {(\mathbb {Z}/2\mathbb {Z})^4} . Further, each of the above occurs infinitely often.

We develop Commutator Theory for congruences of general algebraic systems (henceforth called algebras) assuming only the existence of a ternary term d d such that d ( a , b , b ) [ α , α ] a [ α , α ] d ( b , b , a ) d(a,b,b)[\alpha ,\alpha ]a[\alpha ,\alpha ]d(b,b,a) , whenever α \alpha is a congruence and a α b a\alpha b . Our results apply in particular to congruence modular and n n -permutable varieties, to most locally finite varieties, and to inverse semigroups. We obtain results concerning permutability of congruences, abelian and solvable congruences, connections between congruence identities and commutator identities. We show that many lattices cannot be embedded in the congruence lattice of algebras satisfying our hypothesis. For other lattices, some intervals are forced to be abelian, and others are forced to be nonabelian. We give simplified proofs of some results about the commutator in modular varieties, and generalize some of them to single algebras having a modular congruence lattice.

A class V \mathcal {V} of algebras of the same type is said to be finitely decidable iff the first order theory of the class of finite members of V \mathcal {V} is decidable. Let V \mathcal {V} be a congruence modular variety. In this paper we prove that if V \mathcal {V} is finitely decidable, then the following hold. (1) Each finitely generated subvariety of V \mathcal {V} has a finite bound on the cardinality of its subdirectly irreducible members. (2) Solvable congruences in any locally finite member of V \mathcal {V} are abelian. In addition we obtain various necessary conditions on the congruence lattices of finite subdirectly irreducible algebras in V \mathcal {V} .

A kernel-trace description of right congruences on an inverse semigroup is developed. It is shown that the trace mapping is a complete ∩ \cap homomorphism but not a ∨ \vee -homomorphism. However, the trace classes are intervals in the complete lattice of right congruences. In contrast, each kernel class has a maximum element, namely the principal right congruence on the kernel, but in general there is no minimum element in a kernel class. The kernel mapping preserves neither intersections nor joins. The set of axioms presented in [7] for right kernel systems is reviewed. A new set of axioms is obtained as a consequence of the fact that a right congruence is the intersection of the principal right congruences on the idempotent classes. Finally, it is shown that even though a congruence on a regular semigroup is the intersection of the principal congruences on the idempotent classes, the situation is not the same for right congruences on a regular semigroup. Right congruences on a regular, even orthodox, semigroup are not, in general, determined by their idempotent classes.

The lattice of all complete congruence relations of a complete lattice is itself a complete lattice. In this paper, we characterize this lattice as a complete lattice. In other words, for a complete lattice L L , we construct a complete lattice K K such that L L is isomorphic to the lattice of complete congruence relations of K K . Regarding K K as an infinitary algebra, this result strengthens the characterization theorem of congruence lattices of infinitary algebras of G. Grätzer and W. A. Lampe. In addition, we show how to construct K K so that it will also have a prescribed automorphism group.

A congruence ρ \rho on an inverse semigroup S S is determined uniquely by its kernel and its trace. Denoting by ρ min {\rho ^{\min }} and ρ min {\rho _{\min }} the least congruence on S S having the same kernel and the same trace as ρ \rho , respectively, and denoting by ω \omega the universal congruence on S S , we consider the sequence ω \omega , ω min {\omega ^{\min }} , ω min {\omega _{\min }} , ( ω min ) min {({\omega ^{\min }})_{\min }} , ( ω min ) min … {({\omega _{\min }})^{\min }} \ldots . These congruences, together with the intersections of corresponding pairs, form a sublattice of the lattice of all congruences on S S . We study the properties of these congruences and establish several properties of the quasivarieties of inverse semigroups induced by them.

Let S S be a regular semigroup and let ρ \rho be a congruence relation on S S . The kernel of ρ \rho , in notation ker ⁡ ρ \ker \rho , is the union of the idempotent ρ \rho -classes. The trace of ρ \rho , in notation tr ρ \operatorname {tr}\,\rho , is the restriction of ρ \rho to the set of idempotents of S S . The pair ( ker ⁡ ρ , tr ρ ) (\ker \rho ,\operatorname {tr}\,\rho ) is said to be the congruence pair associated with ρ \rho . Congruence pairs can be characterized abstractly, and it turns out that a congruence is uniquely determined by its associated congruence pair. The triple ( ( ρ ∨ L ) / L , ker ⁡ ρ , ( ρ ∨ R ) / R ) ((\rho \vee \mathcal {L})/\mathcal {L},\ker \rho ,(\rho \vee \mathcal {R})/\mathcal {R}) is said to be the congruence triple associated with ρ \rho . Congruence triples can be characterized abstractly and again a congruence relation is uniquely determined by its associated triple. The consideration of the parameters which appear in the above-mentioned representations of congruence relations gives insight into the structure of the congruence lattice of S S . For congruence relations ρ \rho and θ \theta , put ρ T l θ [ ρ T r θ , ρ T θ ] \rho {T_l}\theta \;[\rho {T_r}\theta ,\rho T\theta ] if and only if ρ ∨ L = θ ∨ L [ ρ ∨ R = θ ∨ R , tr ⁡ ρ = tr ⁡ θ ] \rho \vee \mathcal {L} = \theta \vee \mathcal {L}\;[\rho \vee \mathcal {R} = \theta \vee \mathcal {R},\operatorname {tr}\rho = \operatorname {tr}\theta ] . Then T l , T r {T_l},{T_r} and T T are complete congruences on the congruence lattice of S S and T = T l ∩ T r T = {T_l} \cap {T_r} .

There are several generalizations to universal algebras of the notion "The groupA\mathfrak {A}is an extension of the groupB\mathfrak {B}by the groupC\mathfrak {C}". In this paper we study three such generalizations and the corresponding products of classes of algebraic systems. Various results are presented. One such theorem characterizes the weakly congruence regular varieties admitting extensions of a particular sort. Another result gives, under a weak congruence permutability condition, an equational basis for the variety obtained by applying one such product to two other varieties.

This paper investigates various radicals and radical congruences of a semigroup. A strongly prime ideal is defined. It is shown that the nil radical of a semigroup is the intersection of all strongly prime ideals of the semigroup. Furthermore, a semigroup with zero element is nil if and only if it has no strongly prime ideals. We investigate the question of when the left and right radical congruence relations of various radicals are equal. Some theorems analogous to theorems concerning the radicals of rings are also proved.

Let Γ ( 8 ) \Gamma (8) denote the principal congruence subgroup of level 8 and let Γ ( 16 , 32 ) \Gamma (16,32) denote the subgroup of Γ ( 16 ) \Gamma (16) satisfying c d ≡ a b ≡ 0 ( mod 32 ) cd \equiv ab \equiv 0\;(\bmod 32) . We are dealing only with the elliptic modular case. Consider the spaces of cusp forms of weight 2 (differentials of the first kind) with respect to these groups. It is proved that these spaces are generated by certain monomials of theta constants of degree 4.

Absolute subretracts and weak injectives in congruence modular varieties of universal algebras are investigated by focusing attention on the directly indecomposibles. The proofs rely on a congruence modular version of generalized direct products (direct products with amalgamation) and on the generalized Jónsson Lemma for congruence modular varieties. The results have immediate application to varieties of groups or rings.

In this paper we generalize a result of Tamura on .S-indecomposable semigroups. Based on this, the concept of a minimal sequence between two points, and from a point to another, is introduced. The relationship between two minimal sequences between the same points is studied. The rank of a semigroup S is defined to be the supremum of the lengths of the minimal sequences between points in S. The semirank of a semigroup S is defined to be the supremum of the lengths of the minimal sequences from a point to another in S. Rank and semirank are further studied. Introduction. Semilattice decompositions of semigroups were first defined and studied by Clifford [1]. Since then several people have worked on this topic, notably Tamura (5H{9]. The author's work on the subject can be found in (3], (4]. In this paper, we start by generalizing a result of Tamura (8] (or (9]) on Sindecomposable semigroups. Based on this, the concept of a minimal sequence between two points, and from a point to another, is introduced. The relationship between two minimal sequences between the same points is studied. The rank of a semigroup is defined to be the supremum of the lengths of the minimal sequences between points in the semigroup. The semirank of a semigroup is defined to be the supremum of the lengths of the minimal sequences from a point to another in the semigroup. Rank and semirank are further studied. To understand this paper, the reader need only be aware of the first few chapters of Clifford and Preston [2] and Tamura's decomposition theorem. (See any of (5], [6], [8] or (9]. It was rediscovered by Petrich [(0].) I Preliminaries. Throughout, S will denote a semigroup and Z+ the set of positive integers. A congruence a on S is called a semilattice congruence if S/a is a semilattice. S x S is the universal congruence on S. S is S-indecomposable if S x S is the only semilattice congruence on S. Definition. Let a, b E S. Then (1) a I b if and only if b E SaS1. I is transitive and reflexive. (2) is defined as a > b iff a I bi for some i E Z'; let -0 denote --, i.e., = _-. (3) a nI b iff there exists x E S such that a x b. (4)a biffa bforsome Z+. (5) -is defined as a b iff a a; let -? denote -, i.e., -? Received by the editors May 9, 1972 AMS (MOS) subject classfications (1970). Primary 20M10.

A construction of all coextensions of a regular semigroup S S by rectangular bands is obtained. The construction is analogous to Hall’s construction of orthodox semigroups as spined products of bands and inverse semigroups and reduces to that construction when S S is inverse. The results are specialized to provide a construction of the category of all normal coextensions of a regular semigroup.

The notions of A n A_n -maps and C n C_n -forms can be regarded as crude approximations to the concepts of homomorphisms and commutativity, respectively. These approximations are used to study power maps on connected Lie groups and their localizations. For such groups the power map x ↦ x n x \mapsto {x^n} is known to be an A 2 A_2 -map if and only if n n is a solution to a certain quadratic congruence. In this paper, A 3 A_3 -power maps are studied. For the Lie group Sp(l) it is shown that the A 3 A_3 -powers coincide with solutions which are common to the quadratic congruence, mentioned earlier, and another cubic congruence. Other Lie groups, when localized so as to become homotopy commutative, are also shown to have infinitely many A 3 A_3 -powers. The proofs reflect the combinatorial nature of the obstructions involved.

The analytic basis for any projective theory of congruences, in which the lines of the congruence are defined either by a pair of points or a pair of planes, consists of the invariant theory of a completely integrable system of four homogeneous linear partial differential equations with two dependent and two independent variables, two of the equations of the system being of the first order, and two of the second order. If the developables of the congruence are known, this system of differential equations can be written in a very simple form.t But the determination of the developables of a congruence requires the integration of two partial differential equations of the first order, and it seems highly desirable to possess a theory which will be immediately applicable to any congruence, whether its developables can actually be found explicitly or not. The considerations made by G. M. Greent show that the existing theory can actually be modified so as to cover all such cases. However there are various ways in which this can be done and the various methods which might be used are not all equally desirable. Green himself has indicated one such method for the theory of congruences.? But Green only indicated in a general way what was to be done without actually working out a complete theory. The great disadvantage which this particular theory would have, as compared with the one to which we are devoting this paper, is that only the final results would be of interest. The various types of invariants corresponding to the transformations of certain subgroups would

In a real analytic manifold a k dimensional (real) analytic chain is a locally finite sum of integral multiples of chains given by integration over certain k dimensional analytic submanifolds (or strata) of some k dimensional real analytic variety. In this paper, for any integer ν ≥ 2 \nu \geq 2 , the concepts and results of [6] on the continuity of slicing and the intersection theory for analytic chains are fully generalized to the modulo ν \nu congruence classes of such chains.

A simplified proof is given for the theorem characterizing the congruence lattice of a universal algebra.

This paper generalizes the notion of congruence order for metric spaces to k-metric (k-dimensional metric) spaces. The k-congruence order of E k {E_k} with respect to the class of oriented semi k-metric spaces is determined. An example shows that this result is sharp.

where q is a prime of the form Xv + 1, and c, d are integers prime to q. The pioneer work in this field was done by Gauss,t who showed that for X = 3 the determination of the number of solutions could be made to depend on the representation of 4q by the form A2 + 27B2, and for X = 4 on the representation of q by the form A2 + B2. For the general value of X the congruence is intimately related to Jacobi's cyclotomic function, I

The theory of transformations Bk of deforms by flexure of quadricst may be regarded as a later development of the concepts introduced by Sophus Lie in his researches on the transformations of surfaces of constant curvature. The attempt to make a possible extension of the same principles to researches of another class of applicable surfaces leads us naturally to propose a problem of arranging plane elements, orfacettes, in space-which for the sake of clearness we will explain in detail.t A facette consists of a plane and a point of it called the center. We think of a surface S as the totality of its ooI2 facettes f, the planes of the facettes being tangent to S at. their respective centers. We consider associated with each facette f a simple infinity of facettes f', in accordance with any continuous law whatever. We imagine also that in every deformation of S, each facette f and the oo associated facettes f' are carried along as an invariable system. In each configuration of S the associated facettes f' form a triply infinite system, and in general they cannot be arranged into a series of oo1 surfaces S', each consisting of o2 facettesf'. The problem in view consists precisely in determining all the cases for which the above circumstance is true in all deformations of S. Of the general problem thus stated it is easy to indicate an infinity of particular solutions amongst which are immediately evident those in which each of the ool surfaces S' remains constituted always of the same oo2 facettes f'. But, in view of the eventual applications to problems of deformation, it is opportune to limit the problem much more, and to suppose that everv facette f and each of its associated facettes f' has the center of one in the plane of the other. Thus the surface S and each of its transforms S' are always the focal surfaces of the rectilinear congruence formed by the joins of corresponding points.

The class of 0-modular bands of groups is defined by means of a type of modularity condition on the lattice of congruences on a band of groups. The main result characterizes d-modularity as a condition on the multiplication in the band of groups. This result is then applied to the classes of normal bands of groups and orthodox bands of groups. Introduction. The class of 0-modular bands of groups is defined in 11u, where it is shown that 0-modularity is necessary and sufficient for a certain function to embed the lattice of congruences A(S) in a product lattice. In this paper, we investigate some properties of 0-modular bands of groups, and find several conditions equivalent to 0-modularity. The main result gives a condition on the multiplication in a band of groups S which is necessary and sufficient for S to be 0-modular. This result is then applied to the classes of normal bands of groups and orthodox bands of groups, to obtain a very simple characterization of 0-modularity in these classes. 1. Preliminaries. We use the notation and terminology of Clifford and Preston [21, with the following exceptions: x1: the inverse of x in H5, in a band of groups. B(S): the lattice of band congruences on S. M(S): the lattice of idempotent-separating congruences on S. t (x, y)}*: the congruence generated by the relation whose only element is the pair (x, y) (see [2, Theorem 1.81). By a band of groups, we mean a union of groups on which Green's J{-relation is a congruence. Following [12i, a band B will be called normal if axya = ayxa for all a, x, y E B. Theorem 8.2.9 of [61 lists several necessary and sufficient conditions for a band to be normal. A band of groups S is called a normal band of groups if the band S/J{ is normal. It is known that a band of groups S is a normal band of groups if and only if S is a strong semilattice (mapping semigroup) of completely simple semigroups, namely the 93-classes of S. This result Presented to the Society, November 19, 1971 under the title The lattice of congruences on a band of groups; received by the editors April 21, 1972. AMS (MOS) subject classifications (1970). Primary 20M10, 20M15.

be a functional relation between the complex variables z and w. Let us first plot corresponding values of z and w in one and the same plane, and let us then project these points upon the unit sphere by stereographic projection. The lines, which join all pairs of points thus obtained upon the sphere, for a given relation w = F (z), form the congruence in question. If w is not a linear function. of z, every congruence of this class has the following properties. Ia. It is a W-congruence whose focal sheets are distinct, non-degenerate, and non-ruled surfaces. Ib. The focal surfaces are real and have a positive measure of curvature. II. The developables of the congruence determtne isothermally conjugate systems of curves on both sheets of the focal surface. III. The asymptotic curves of both sheets of the focal surface belong to linear complexes. IV. The directrix of the first kind, for every point of either focal sheet, coincides with the directrix of the second kind for the corresponding point of the other focal sheet. Va. The directrix quadrics of both-focal sheets are non-degenerate, and coincide with each other. Vb. Both of these directrix quadrics coincide with the Riemann'sphere. If we prefer, we may replace property II by

1. Professor Birkhoff has shownt that every dynamical problem with two degrees of freedom can be represented by the motioni of a particle under a field of force on a characteristic surface which is either fixed (reversible case) or rotating at uniform velocity anid carrying its field of force with it (irreversible case). In either type of problem the magnitude of the velocity is a function only of the position of the particle on the surface, so that the state of motion at any instant is given by three parameters, two giving the position and onie the direction of motion. We shall for the most part consider the last named parameter as varyinig from 0 to 2 w, regarding opposite directions at a poinlt as distinct; this is essential in the irreversible though not in the reversible case. The three parameters may be regarded as determining a point in a threedimensional manifold. The manifolds thus obtained are considered in this laper from the point of view of analysis situs. It should be noted that these manifolds are not in general, as might be expected, the "product manifolds" whose points are in 11 continnous correspondence with pairs of points, one point of each pair lyinig on the characteristic surface and one on a circle. For example, if the characteristic surface is one-sided the prodnct manifold is one-sided, but the manifold of states of motion is orientable. Only for a stnrface of the coninectivity of the anchor ring are the two manifolds homeomorphic. The underlying reason is the impossibility of a non-singular coordinate system or continuous distribution of vectors on any other surface. Each orbit is represented by a curve in the manifold. These curves are called "stream lines" by Birkhoff (loc. cit.) because of a hydrodynamical analogy. The stream lines constitute a non-singuilar congruence, one througl each point of the manifold. To a periodic orbit corresponds a closed stream line. This relation and the importance of periodic orbits add interest to the problem of finding the properties of the manifolds of states of motion. Some special manifolds of this class are described, from a point of view slightly different from that here adopted, by Birkhoff in the paper above

This paper continues the examination of the structure of pseudocomplemented distributive lattices. First, the Congruence Extension Property is proved. This is then applied to examine properties of the equational classes B n , − 1 ≦ n ≦ ω {\mathcal {B}_n}, - 1 \leqq n \leqq \omega , which is a complete list of all the equational classes of pseudocomplemented distributive lattices (see Part I). The standard semigroups (i.e., the semigroup generated by the operators H, S, and P) are described. The Amalgamation Property is shown to hold iff n ≦ 2 n \leqq 2 or n = ω n = \omega . For 3 ≦ n > ω , B n 3 \leqq n > \omega ,{\mathcal {B}_n} does not satisfy the Amalgamation Property; the deviation is measured by a class Amal ( B n ) ( ⊆ B n ) ({\mathcal {B}_n})( \subseteq {\mathcal {B}_n}) . The finite algebras in Amal ( B n ) ({\mathcal {B}_n}) are determined.

A primary polynomial is one whose first coefficient is 1, and ?D(H) is the number of polynomials in a reduced residue system modulo H. Let M denote the multiplicative semigroup consisting of the primary polynomials in the ring GF[q, x] of polynomials over the finite field of q elements, q being a prime power. An equivalence relation on M is said to be a congruence relation if it is compatible with the semigroup structure of M. If gH denotes the restriction to M of the relation "congruence modulo H" on GF[q,x], then it is clear that H is a congruence relation on M for every H in GF[q, x]. Our aim in this paper is to establish a result similar to Theorem 1.1 for a wider class of congruence relations on M than those of the special form RH. To this end, we have extracted the relevant properties of the relations WH and used these properties to define a class of congruence relations on M which we call the arithmetically distributed relations. The precise definition is given in ?8. Theorem 8.1 of that section states a result for arithmetically distributed relations which is analogous to that stated in Theorem 1.1 for the relations RH. It includes Theorem 1.1 as a special case. The proof given for Theorem 8.1 is similar to that given by Artin for Theorem 1.1 in that certain analytic functions, the L-functions, are introduced and in that the crucial step of the proof lies in showing that these L-functions do not vanish on the line Real(z) = 1. However, the proof differs from that of Artin in

Let y(1) e y(4) and z(1) ... z(4) be interpreted as the homogeneous coordinates of two points Pv and P,. As u and v vary, these points will describe two surfaces, S, and S, (either or both of which may be degenerate), and the line P, P, will generate a congruence whose focal surface consists of the two surfaces S, and S,. Moreover the ruled surfaces of the congruence obtained by equating either u or v to a constant will be its developables.t All congruences whose focal surfaces have two distinct sheets may be studied by this method. The invariants and covariants of a system of form (D) are those functions of the coefficients and variables which are left unchanged (absolutely or except for a factor), when system (D) is subjected to any transformation of the form

In this paper we shall first define the radical of a semigroup S and investigate some of its properties. Just as in the case of rings, one finds that the radical of a semigroup is a quasi-regular two-sided ideal which contains each quasi-regular right ideal of the semigroup and that the radical of a semigroup contains each nil right ideal of the semigroup. If a semigroup with zero satisfies the minimum condition on left ideals and right ideals then its radical is also the left radical of the semigroup. One also finds that if S. is the semigroup of all row-monomial n x n matrices over S then the radical of S,, is the semigroup of all row-monomial n x n matrices over the radical of S. If T is a two-sided ideal of S then the radical of T is the intersection of T and the radical of S. In the latter part of the paper we shall study semisimple semigroups. If s, t E S, let spt provided that, if m is contained in some 0-transitive operand of S, then ms = mt. Thus p is a two-sided congruence called the radical congruence of S. If the radical congruence of S is the identity relation then S is semisimple. Since M is a 0-transitive operand of S if and only if M is a 0-transitive operand of S/p, then it is easily seen that S/p is semisimple. If S is semisimple and T is a two-sided ideal of S, then T is semisimple. Finally, if S contains more than one element, then S is semisimple if and only if S is isomorphic to a subdirect sum of a set of semigroups each of which has a faithful 0-transitive operand.

It is the purpose of this paper to extend some of the ideas related to the transformationt C of nets in projective space of three dimensions to nets in hyperspace. In three dimensions two nets N. and N, are said to be in relation C if the developables of the congruence G of lines joining corresponding points x and y intersect the two sustaining surfaces in those nets, provided however that neither surface is a focal surface of the congruence. If N., and A17 are in relation C, the tangents at x and y to corresponding curves of the nets intersect. We shall say that two nets N., and N,, in space Sn of n dimensions are in relation C if the two sustaining surfaces Sx and S2, are such that corresponding tangent planes intersect in a line, and if the developable surfaces of the congruence G of lines g joining corresponding points x and y intersect the two surfaces in the nets. It is to be understood that the two sustaining surfaces are not the focal surfaces of G. Not all nets Nx in Sn for n > 4 can have a C transform N,. A net in Sn which permits of having a C transform will be called a C net. We derive necessary and sufficient conditions that a non-conjugate net be a C net. Another geometrical interpretation is given for two covariant points found by Bompiani.4 Let Sx and S, be two surfaces in the same space Sn of n dimensions and with their points in one-to-one point correspondence. Let the parametric equations of these surfaces be

In this paper all subdirectly irreducible pseudocomplemented distributive lattices are found. This result is used to establish a Stone-like representation theorem conjectured by G. Grätzer and to find all equational subclasses of the class of pseudocomplemented distributive lattices.

Let there be given a surface S in euclidean space of n>3 dimensions. Suppose that through each point x of S there passes a line g of a congruence G. The developables of G intersect S in a net of curves N. We have called such a net a C net. t Let S be another surface in the same space S, in one-to-one point correspondence with S, corresponding points lying on the lines g of G. The developables of G intersect S in a C net of curves N. The two nets N and N are said to be in relatio4t C. The tangent planes to S and S intersect in a line h. If the points of h are each equidistant from the corresponding points of S and S, we shall say that the nets N and N are in relation E. We propose in this paper to develop a theory of the relations defined above which is independent of the dimension of the space S, for n > 3. Let the coordinates of the point x on S be xi, x2, **, x, the coordinates of the corresponding point x on S be x, x , x, and the direction cosines of the line g joining them be X1, X2, ** Xn where

Introduction. It is the purpose of this paper to establish certain transformations for any non-developable surface that bears a family of asymptotic curves and is immersed in a space of n dimensions Sn (n >3). All surfaces mentioned hereafter will belong to Sn The ambient of the osculating planes at a point of a surface to all of the curves on the surface that go through the point is a space of not more than five dimensions.t The class of all surfaces in Sn for which the ambient at all points is a space of four dimensions is divided into two subclasses. One of these subclasses is composed of all surfaces in Sn that bear each a conjugate net of curves, while the other is composed of all surfaces in S,, that sustain each a family of asymptotic curves. In the classical transformations for a surface bearing a conjugate net, the two congruences of lines tangent to the curves of the net have played basic r6les. We shall assign a similar role to the c 2 lines tangent to the asymptotic curves of a family. Although the congruence of these lines contains only a one parameter family of developable surfaces, it will be defined as a parabolic congruence. To facilitate discussion, a terminology for certain geometric relations is introduced.