Partial Algebras Research Articles

Partial combinatory algebra and generalized numberings

Generalized numberings are an extension of Ershov's notion of numbering, based on partial combinatory algebra (pca) instead of the natural numbers. We study various algebraic properties of generalized numberings, relating properties of the numbering to properties of the pca. As in the lambda calculus, extensionality is a key notion here.

The plethystic inverse of the odd Lie representations

The Frobenius characteristic of L i e n Lie_n , the representation of the symmetric group S n S_n afforded by the multilinear part of the free Lie algebra, is known to satisfy many interesting plethystic identities. In this paper we prove a conjecture of Richard Stanley establishing the plethystic inverse of the sum ∑ n ≥ 0 L i e 2 n + 1 \sum _{n\geq 0} Lie_{2n+1} of the odd Lie characteristics. We obtain an apparently new plethystic decomposition of the regular representation of S n S_n in terms of irreducibles indexed by hooks, and the Lie representations. We also determine the plethystic inverse of the alternating sum of the odd Lie characteristics.

Partial actions of a Hopf algebra on its base field and the corresponding partial smash product algebra

In this paper, we introduce the concept of a [Formula: see text]-Hopf algebra as a Hopf algebra obtained as the partial smash product algebra of a Hopf algebra and its base field, and show that every Hopf algebra is a [Formula: see text]-Hopf algebra. Moreover, a method to compute partial actions of a given Hopf algebra on its base field is developed and, as an application, we exhibit all partial actions of such type for some families of Hopf algebras.

Totalising Partial Algebras

We will examine totalising a partial operation in a general algebra by using an absorbtive element, bottom, such as an error flag. We then focus on the simplest example of a partial operation, namely subtraction on the natural numbers: n - m is undefined whenever n < m. We examine the use of bottom in algebraic structures for the natural numbers, especially semigroups and semirings. We axiomatise this totalisation process and introduce the algebraic concept of a team, being an additive cancellative semigroup with totalised subtraction. Also, with the natural numbers in mind, we introduce the property of being generated by an iterative function, which we call a splinter. We prove a number of theorems about the algebraic specification of datatypes of natural numbers.

Tamagawa number divisibility of centralL-values of twists of the Fermat elliptic curve

Given any integer N>1 prime to 3, we denote by C N the elliptic curve x 3 +y 3 =N. We first study the 3-adic valuation of the algebraic part of the value of the Hasse–Weil L-function L(C N ,s) of C N over ℚ at s=1, and we exhibit a relation between the 3-part of its Tate–Shafarevich group and the number of distinct prime divisors of N which are inert in the imaginary quadratic field K=ℚ(-3). In the case where L(C N ,1)≠0 and N is a product of split primes in K, we show that the order of the Tate–Shafarevich group as predicted by the conjecture of Birch and Swinnerton-Dyer is a perfect square.

Conformal bootstrap with reinforcement learning

We introduce the use of reinforcement-learning (RL) techniques to the conformal-bootstrap programme. We demonstrate that suitable soft Actor-Critic RL algorithms can perform efficient, relatively cheap high-dimensional searches in the space of scaling dimensions and OPE-squared coefficients that produce sensible results for tens of CFT data from a single crossing equation. In this paper we test this approach in well-known 2D CFTs, with particular focus on the Ising and tri-critical Ising models and the free compactified boson CFT. We present results of as high as 36-dimensional searches, whose sole input is the expected number of operators per spin in a truncation of the conformal-block decomposition of the crossing equations. Our study of 2D CFTs uses only the global $so(2,2)$ part of the conformal algebra, and our methods are equally applicable to higher-dimensional CFTs. When combined with other, already available, numerical and analytical methods, we expect our approach to yield an exciting new window into the non-perturbative structure of arbitrary (unitary or non-unitary) CFTs.

Partial Menger algebras and their weakly isomorphic representation

As generalization of semigroups, Karl Menger introduced in the 1940th algebras of multiplace operations. Such algebras satisfy the superassociative law, a generalization of the associative law. Menger algebras are defined as models of this superassociative law. Cayley’s theorem for semigroups says that any model of the associative law is isomorphic to a transformation semigroup. R. M. Dicker proved in 1963 that every Menger algebra of rank [Formula: see text] is isomorphic to a Menger algebra of [Formula: see text]-ary operations on some set. The composition of terms in which each variable occurs at most [Formula: see text]-times, so-called [Formula: see text]-terms, leads to partial algebras where the superassociative law is satisfied as a weak identity. In this paper, we introduce the concepts of a partial Menger algebra, a unitary partial Menger algebra and of a generalized partial Menger algebra. We prove that [Formula: see text]-terms of some type form a generalized partial Menger algebra with infinitely many nullary operations. Using weak identities and weak isomorphisms, Dicker’s result will be extended to partial Menger algebras and to unitary partial Menger algebras.

Diagonal decompositions of shapes and their algebras

AbstractThe formal approach to shapes and their algebras, as it appears in shape grammar theory, has been reviewed. It starts with geometric elements and their partial algebras, continues to shapes, their algebras, and boundaries, as well as algebras that calculate with shapes and their boundaries. There is a number of new concepts introduced along the way. These include diagonal decompositions and their algebras which simplify calculations with shapes, b-paired diagonal decompositions which extend calculations with shapes and their boundaries from diagonal shapes only to all shapes, andm-order boundaries which extend the concept of shape boundaries and allow for calculations with multiple representations of shapes. It also shows that algebras of shapes are infinite direct sums of diagonal algebras.

A Study on Soft Multi-Granulation Rough Sets and Their Applications

Rough set (RS) and soft set (SS) theories are two successful mathematical approaches to dealing with uncertainty in data analysis. The classical soft rough set (SRS) theory proposed by Feng et al. [8] offers a formal theoretical framework for solving the uncertainty under a single granulation environment. However, it is essential to note that the SRS theory cannot be applied in the context of multi-granulation in the real world. To address this issue, in this paper, we introduce the idea of soft multi-granulation RS (SMGRS) model based on two soft binary relations (S-BRs). Axiomatic operations, lower soft rough approximation space (lower SRA-space) and upper soft rough approximation space (upper SRA-space), are defined through after sets of soft relations. After that, the concept of SMGRSs is applied to a significant part of commutative algebra, group theory. In this respect, the primitive notions of SRA-spaces are defined with the help of two normal soft groups (NSGs). In groups, several important structural properties related to SMGRS are investigated in detail with illustrative examples. It is shown that SMGRS in groups may be influential in decision-making (DM) by some numerical examples. To demonstrate the flexibility, superiority, and effectiveness of the suggested technique, some comparative examples are given with some existing methods.

The Algebraic Sets in the General Mathematics and Teaching Them

Most of the pictures in general mathematics are algebraic sets. Indeed, even the first figures taught in class 1 of elementary school are already algebraic sets or part of algebraic sets, such as lines and segments. Therefore, knowing with certainty the properties of algebraic sets is very important for good teaching of high school mathematics, and it is essential to teach them better. To give suggestions and help teachers teach mathematics more effectively, in this report, we will present the Zariski topology, some of their most important properties and the methods to teach algebraic sets.

Lie Algebras Arising from Nichols Algebras of Diagonal Type

Abstract Let $\mathcal{B}_{\mathfrak{q}}$ be a finite-dimensional Nichols algebra of diagonal type with braiding matrix $\mathfrak{q}$, $\mathcal{L}_{\mathfrak{q}}$ be the corresponding Lusztig algebra as in [ 4], and $\operatorname{Fr}_{\mathfrak{q}}: \mathcal{L}_{\mathfrak{q}} \to U(\mathfrak{n}^{\mathfrak{q}})$ be the corresponding quantum Frobenius map as in [ 5]. We prove that the finite-dimensional Lie algebra $\mathfrak{n}^{\mathfrak{q}}$ is either 0 or the positive part of a semisimple Lie algebra $\mathfrak{g}^{\mathfrak{q}}$, which is determined for each $\mathfrak{q}$ in the list of [ 25].

Truncated cluster algebras and Feynman integrals with algebraic letters

We propose that the symbol alphabet for classes of planar, dual-conformal-invariant Feynman integrals can be obtained as truncated cluster algebras purely from their kinematics, which correspond to boundaries of (compactifications of) G+(4, n)/T for the n-particle massless kinematics. For one-, two-, three-mass-easy hexagon kinematics with n = 7, 8, 9, we find finite cluster algebras D4, D5 and D6 respectively, in accordance with previous result on alphabets of these integrals. As the main example, we consider hexagon kinematics with two massive corners on opposite sides and find a truncated affine D4 cluster algebra whose polytopal realization is a co-dimension 4 boundary of that of G+(4, 8)/T with 39 facets; the normal vectors for 38 of them correspond to g-vectors and the remaining one gives a limit ray, which yields an alphabet of 38 rational letters and 5 algebraic ones with the unique four-mass-box square root. We construct the space of integrable symbols with this alphabet and physical first-entry conditions, whose dimension can be reduced using conditions from a truncated version of cluster adjacency. Already at weight 4, by imposing last-entry conditions inspired by the n = 8 double-pentagon integral, we are able to uniquely determine an integrable symbol that gives the algebraic part of the most generic double-pentagon integral. Finally, we locate in the space the n = 8 double-pentagon ladder integrals up to four loops using differential equations derived from Wilson-loop d log forms, and we find a remarkable pattern about the appearance of algebraic letters.

Isomorphisms and derivations of partial flag incidence algebras

Let [Formula: see text] and [Formula: see text] be finite posets and [Formula: see text] a commutative unital ring. In the case where [Formula: see text] is indecomposable, we prove that the [Formula: see text]-linear isomorphisms between partial flag incidence algebras [Formula: see text] and [Formula: see text] are exactly those induced by poset isomorphisms between [Formula: see text] and [Formula: see text]. We also show that the [Formula: see text]-linear derivations of [Formula: see text] are trivial.

Similarity-based equational inference in physics

The authors automate partial computer algebra derivations from a contemporary result in physics, propose a method for converting handwritten derivations into computer algebra, and describe a dataset produced via this approach.

The Algebra of Non-deterministic Programs: Demonic Operations, Orders and Axioms

Abstract Demonic composition, demonic refinement and demonic union are alternatives to the usual ‘angelic’ composition, angelic refinement (inclusion) and angelic (usual) union defined on binary relations. We first motivate both the angelic and the demonic via an analysis of the behaviour of non-deterministic programs, with the angelic associated with partial correctness and demonic with total correctness, both cases emerging from a richer algebraic model of non-deterministic programs incorporating both aspects. Zareckiĭ has shown that the isomorphism class of algebras of binary relations under angelic composition and inclusion is finitely axiomatized as the class of ordered semigroups. The proof can be used to establish that the same axiomatization applies to binary relations under demonic composition and refinement, and a further modification of the proof can be used to incorporate a zero element representing the empty relation in the angelic case and the full relation in the demonic case. For the signature of angelic composition and union, it is known that no finite axiomatization exists, and we show the analogous result for demonic composition and demonic union by showing that the same axiomatization holds for both. We show that the isomorphism class of algebras of binary relations with the ‘mixed’ signature of demonic composition and angelic inclusion has no finite axiomatization. As a contrast, we show that the isomorphism class of partial algebras of binary relations with the partial operation of constellation product and inclusion (also a ‘mixed’ signature) is finitely axiomatizable.

The canonical formulation of E6(6) exceptional field theory

We investigate the canonical formulation of the (bosonic) E6(6) exceptional field theory. The explicit non-integral (not manifestly gauge invariant) topological term of E6(6) exceptional field theory is constructed and we consider the canonical formulation of a model theory based on the topological two-form kinetic term. Furthermore we construct the canonical momenta and the canonical Hamiltonian of the full bosonic E6(6) exceptional field theory. Most of the canonical gauge transformations and some parts of the canonical constraint algebra are calculated. Moreover we discuss how to translate the results canonically into the generalised vielbein formulation. We comment on the possible existence of generalised Ashtekar variables.

On Zagier-Hoffman's conjectures in positive characteristic

We study Todd-Thakur's analogues of Zagier-Hoffman's conjectures in positive characteristic. These conjectures predict the dimension and an explicit basis $\mathcal{T}_w$ of the span of characteristic $p$ multiple zeta values of fixed weight $w$ which were introduced by Thakur as analogues of classical multiple zeta values of Euler. In the present paper we first establish the algebraic part of these conjectures which states that the span of characteristic $p$ multiple zeta values of weight $w$ is generated by the set $\mathcal{T}_w$. As a consequence, we obtain upper bounds for the dimension. This is the analogue of Brown's theorem and also those of Deligne-Goncharov and Terasoma. We then prove two results towards the transcendental part of these conjectures. First, we establish the linear independence for a large subset of $\mathcal{T}_w$ and yield lower bounds for the dimension. Second, for small weights we prove the linear independence for the whole set $\mathcal{T}_w$ and completely solve Zagier-Hoffman's conjectures in positive characteristic. Our key tool is the Anderson-Brownawell-Papanikolas criterion for linear independence in positive characteristic.

$$\frac{1}{2}$$-derivations of Lie algebras and transposed Poisson algebras

A relation between $\frac{1}{2}$-derivations of Lie algebras and transposed Poisson algebras was established. Some non-trivial transposed Poisson algebras with a certain Lie algebra (Witt algebra, algebra $\mathcal{W}(a,-1)$, thin Lie algebra and solvable Lie algebra with abelian nilpotent radical) were constructed. In particular, we constructed an example of the transposed Poisson algebra with associative and Lie parts isomorphic to the Laurent polynomials and the Witt algebra. On the other side, it was proven that there are no non-trivial transposed Poisson algebras with Lie algebra part isomorphic to a semisimple finite-dimensional algebra, a simple finite-dimensional superalgebra, the Virasoro algebra, $N=1$ and $N=2$ superconformal algebras, or a semisimple finite-dimensional $n$-Lie algebra.

ORDINAL ANALYSIS OF PARTIAL COMBINATORY ALGEBRAS

AbstractFor every partial combinatory algebra (pca), we define a hierarchy of extensionality relations using ordinals. We investigate the closure ordinals of pca’s, i.e., the smallest ordinals where these relations become equal. We show that the closure ordinal of Kleene’s first model is ${\omega _1^{\textit {CK}}}$ and that the closure ordinal of Kleene’s second model is $\omega _1$ . We calculate the exact complexities of the extensionality relations in Kleene’s first model, showing that they exhaust the hyperarithmetical hierarchy. We also discuss embeddings of pca’s.

Feynman Integrals and Scattering Amplitudes from Wilson Loops.

We study Feynman integrals and scattering amplitudes in N=4 super-Yang-Mills theory by exploiting the duality with null polygonal Wilson loops. As the main application, we compute for the first time the symbols of the general double pentagon integrals, which give the finite part of two-loop maximally helicity violating (MHV) amplitudes and finite components of next-to-MHV (NMHV) amplitudes to all multiplicities. The rational parts of the symbol consist of 164 letters, while the algebraic part contains 96 algebraic letters and cancel in MHV amplitudes and NMHV components which are free of square roots.