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A variety of algebras is called Cross if it is finitely based, finitely generated, and has finitely many subvarieties. In this paper, we classify all Cross varieties of aperiodic monoids with commuting idempotents. [ABSTRACT FROM AUTHOR]

This paper determines relations between two notions concerning monoids: factorability structure, introduced to simplify the bar complex; and quadratic normalization, introduced to generalize quadratic rewriting systems and normalizations arising from Garside families. Factorable monoids are characterized in the axiomatic setting of quadratic normalizations. Additionally, quadratic normalizations of class 4 , 3 are characterized in terms of factorability structures and a condition ensuring the termination of the associated rewriting system. [ABSTRACT FROM AUTHOR]

An algebra that generates a variety with uncountably many subvarieties is said to be of type 2 ℵ 0 . We show that the Rees quotient monoid M(aabb) of order ten is of type 2 ℵ 0 , thereby affirmatively answering a recent question of Glasson. As a corollary, we exhibit a new example of type 2 ℵ 0 monoid of order six, which turns out to be the minimal possible cardinality and the first of its kind that is finitely based. [ABSTRACT FROM AUTHOR]

Let X be a group, Y be a monoid, K be a submonoid of Y and n (Y) : = 2 Y \ { ∅ } . In the paper we prove that under some additional assumptions set-valued K-homomorphism G : S → n (Y) defined on a subgroup S of X can be extended to a set-valued K-homomorphism F : X → n (Y) . [ABSTRACT FROM AUTHOR]

After fixing a commutative ring with unit R, we present the definition of adequate category and consider the category of R-linear functors from an adequate category to the category of R-modules. We endow this category of functors with a monoidal structure and study monoids (generalized Green functors) over it. For one of these generalized Green functors, we define two new monoids, its commutant and its center, and study some of their properties and relations between them. This work generalizes the article [3]. [ABSTRACT FROM AUTHOR]

It is shown that the Hecke–Kiselman monoid HK Θ associated to a finite oriented graph Θ satisfies a semigroup identity if and only if HK Θ does not have free noncommutative subsemigroups. It follows that this happens exactly when the semigroup algebra K [ HK Θ ] over a field K satisfies a polynomial identity. The latter is equivalent to a condition expressed in terms of the graph Θ . The proof allows to derive concrete identities satisfied by such monoids HK Θ . [ABSTRACT FROM AUTHOR]

We develop the idempotent theory for algebras over a class of semigroups called left regular bands of groups (LRBGs), which simultaneously generalize group algebras of finite groups and left regular band (LRB) algebras. Our techniques weave together the representation theory of finite groups and LRBs, opening the door for a systematic study of LRBGs in an analogous way to LRBs. We apply our results to construct complete systems of primitive orthogonal idempotents in the Mantaci–Reutenauer algebra MR n [ G ] associated to any finite group G. When G is abelian, we give closed form expressions for these idempotents, and when G is the cyclic group of order two, we prove that they recover idempotents introduced by Vazirani. [ABSTRACT FROM AUTHOR]

In this paper, we give a sufficient condition under which an involution monoid generates a variety with continuum many subvarieties. According to this result, several involution J -trivial monoids are shown to generate varieties with continuum many subvarieties. These examples include Rees quotients of free involution monoids, Lee monoids with involution, and Straubing monoids with involution. [ABSTRACT FROM AUTHOR]

Equivariant monoids are very important objects in many branches of mathematics: they combine the notion of multiplication and the concept of a group action. In this paper we will construct categories which encode the structure borne by monoids with a group action by combining the theory of product and permutation categories (PROPs) and product and braid categories (PROBs) with the theory of crossed simplicial groups. PROPs and PROBs are categories used to encode structures borne by objects in symmetric and braided monoidal categories respectively, whilst crossed simplicial groups are categories which encode a unital, associative multiplication and a compatible group action. We will produce PROPs and PROBs whose categories of algebras are equivalent to the categories of monoids, comonoids and bimonoids with group action using extensions of the symmetric and braid crossed simplicial groups. We will extend this theory to balanced braided monoidal categories using the ribbon braid crossed simplicial group. Finally, we will use the hyperoctahedral crossed simplicial group to encode the structure of an involutive monoid with a compatible group action. [ABSTRACT FROM AUTHOR]

Basic information about figurative numbers is provided. Then, information about isosceles triangular numbers, one of the two-dimensional figurative numbers, is given. It also includes information about algebraic structures and their definitions. Additionally, a binary operation that includes k -isosceles triangular numbers is presented, and the study investigates whether the algebraic structures defined with this operation form a groupoid or semigroup. Also, two examples are given that satisfy the results at the end of the paper. [ABSTRACT FROM AUTHOR]

Let X be a class of metric spaces and let Px be the set of all f: [0, oo) [0, oo) preserving X, i.e., (/, f o p) e X whenever (/, p) e X. For arbitrary subset A of the set of all metric preserving functions, we show that the equality Px = A has a solution if A is a monoid with respect to the operation of function composition. In particular, for the set SI of all amenable subadditive increasing functions, there is a class X of metric spaces such that Px = SI holds. 2020 Mathematics Subject Classification: Primary 26A30, Secondary 54E35, 20M20 [ABSTRACT FROM AUTHOR]

A notion of normal submonoid of a monoid M is introduced that generalizes the normal subgroups of a group. When ordered by inclusion, the set $\mathsf {NorSub}(M)$ of normal submonoids of M is a complete lattice. Joins are explicitly described and the lattice is computed for the finite full transformation monoids $T_n$ , $n\geq ~1$. It is also shown that $\mathsf {NorSub}(M)$ is modular for a specific family of commutative monoids, including all Krull monoids, and that it, as a join semilattice, embeds isomorphically onto a join subsemilattice of the lattice $\mathsf {Cong}(M)$ of congruences on M. This leads to a new strategy for computing $\mathsf {Cong}(M)$ consisting of computing $\mathsf {NorSub}(M)$ and the so-called unital congruences on the quotients of M modulo its normal submonoids. This provides a new perspective on Malcev's computation of the congruences on $T_n$. [ABSTRACT FROM AUTHOR]

In this paper we provide an account of the Todd–Coxeter algorithm for computing congruences on semigroups and monoids. We also give a novel description of an analogue for semigroups of the so-called Felsch strategy from the Todd–Coxeter algorithm for groups. [ABSTRACT FROM AUTHOR]

Let X be an Abelian group, Y be a commutative monoid, K ⊂Y be a submonoid and F : X → 2Y \ {∅} be a set-valued map. Under some additional assumptions on ideals ℐ1 in X and ℐ2 in X2, we prove that if F is ℐ2-almost everywhere K-additive, then there exists a unique up to K K-additive set-valued map G : X → 2Y \{∅} such that F = G ℐ1-almost everywhere in X. Our considerations refers to the well known de Bruijn’s result [1].

Equivalence relations or, more general, quasiorders (i.e., reflexive and transitive binary relations) ϱ have the property that an n-ary operation f preserves ϱ , i.e., f is a polymorphism of ϱ , if and only if each translation (i.e., unary polynomial function obtained from f by substituting constants) preserves ϱ , i.e., it is an endomorphism of ϱ. We introduce a wider class of relations—called generalized quasiorders—of arbitrary arities with the same property. With these generalized quasiorders we can characterize all algebras whose clone of term operations is determined by its translations by the above property, what generalizes affine complete algebras. The results are based on the characterization of so-called u-closed monoids (i.e., the unary parts of clones with the above property) as Galois closures of the Galois connection End – gQuord , i.e., as endomorphism monoids of generalized quasiorders. The minimal u-closed monoids are described explicitly. [ABSTRACT FROM AUTHOR]

Factorizations of monoids are studied. Two necessary and sufficient conditions in terms of the so-called descent 1-cocycles for a monoid to be factorized through two submonoids are found. A full classification of those factorizations of a monoid whose one factor is a subgroup of the monoid is obtained. The relationship between monoid factorizations and non-abelian cohomology of monoids is analyzed. Some applications of semi-direct product of monoids are given. [ABSTRACT FROM AUTHOR]

A variety is finitely universal if its lattice of subvarieties contains an isomorphic copy of every finite lattice. We show that the 6-element Brandt monoid generates a finitely universal variety of monoids and, by the previous results, it is the smallest generator for a monoid variety with this property. It is also deduced that the join of two Cross varieties of monoids can be finitely universal. In particular, we exhibit a finitely universal variety of monoids with uncountably many subvarieties which is the join of two Cross varieties of monoids whose lattices of subvarieties are the 6-element and the 7-element chains, respectively. [ABSTRACT FROM AUTHOR]

An example of a nonfinitely based involution monoid of order five has recently been discovered. We confirm that this example is, up to isomorphism, the unique smallest among all involution monoids. [ABSTRACT FROM AUTHOR]

In this paper, we define Uncertain Measurable S-acts and morphisms between two Uncertain Measurable S-acts on monoids. Next, we construct the new category, namely Uncertain Meas Act-S, of these objects and morphisms. We prove that the category of these new objects is closed under product, coproduct, pushout and pullback. Also, we show that the category of Uncertain Measurable S-acts is closed under equalizer and coequalizer. [ABSTRACT FROM AUTHOR]

Let X be a class of metric spaces and let PX be the set of all f : [0, ∞) → [0, ∞) preserving X, i.e., (Y, f ∘ ρ) ∈ X whenever (Y, ρ) ∈ X. For arbitrary subset A of the set of all metric preserving functions, we show that the equality PX = A has a solution if A is a monoid with respect to the operation of function composition. In particular, for the set SI of all amenable subadditive increasing functions, there is a class X of metric spaces such that PX = SI holds.2020 Mathematics Subject ClassificationPrimary 26A30, Secondary 54E35, 20M20

For a semigroup $S$, the covering number of $S$ with respect to semigroups, $\sigma_s(S)$, is the minimum number of proper subsemigroups of $S$ whose union is $S$. This article investigates covering numbers of semigroups and analogously defined covering numbers of inverse semigroups and monoids. Our three main theorems give a complete description of the covering number of finite semigroups, finite inverse semigroups, and monoids (modulo groups and infinite semigroups). For a finite semigroup that is neither monogenic nor a group, its covering number is two. For all $n\geq 2$, there exists an inverse semigroup with covering number $n$, similar to the case of loops. Finally, a monoid that is neither a group nor a semigroup with an identity adjoined has covering number two as well.

A graph is a core or unretractive if all its endomorphisms are automorphisms. Well-known examples of cores include the Petersen graph and the graph of the dodecahedron—both generalized Petersen graphs. We characterize the generalized Petersen graphs that are cores. A simple characterization of endomorphism-transitive generalized Petersen graphs follows. This extends the characterization of vertex-transitive generalized Petersen graphs due to Frucht, Graver, and Watkins and solves a problem of Fan and Xie. Moreover, we study generalized Petersen graphs that are (underlying graphs of) Cayley graphs of monoids. We show that this is the case for the Petersen graph, answering a recent mathoverflow question, for the Desargues graphs, and for the Dodecahedron—answering a question of Knauer and Knauer. Moreover, we characterize the infinite family of generalized Petersen graphs that are Cayley graphs of a monoid with generating connection set of size two. This extends Nedela and Škoviera's characterization of generalized Petersen graphs that are group Cayley graphs and complements results of Hao, Gao, and Luo. [ABSTRACT FROM AUTHOR]

We study the endomorphisms and derivations of an infinite-dimensional cyclic Leibniz algebra. Among others it was found that if L is a cyclic infinite-dimensional Leibniz algebra over a field F , then the group of all automorphisms of L is isomorphic to a multiplicative group of the field F. The description of an algebra of derivations of a cyclic infinite-dimensional Leibniz algebra has been obtained. [ABSTRACT FROM AUTHOR]

Let M, N be monoids, and PSh(M), PSh(N) their respective categories of right actions on sets. In this paper, we systematically investigate correspondences between properties of geometric morphisms PSh(M) → PSh(N) and properties of the semigroup homomorphisms M → N or flat-left-N-right-M-sets inducing them. More specifically, we consider properties of geometric morphisms featuring in factorization systems, namely: surjections, inclusions, localic morphisms, hyperconnected morphisms, terminal-connected morphisms, ´etale morphisms, pure morphisms and complete spreads. We end with an application of topos-theoretic Galois theory to the special case of toposes of the form PSh(M). [ABSTRACT FROM AUTHOR]

The article presents a novel approach to counting letters in spanish words or phrases using the fingers of the right hand. An algebraic model based on matrices represents the counting process, ensuring that each finger and letter are associated. It is demonstrated that the count is repeated once or five times, leaving no fingers or letters unassociated. Letter matrices are introduced as a representation associated with counting, forming a monoid structure. This study delves into the counting of letters and its relationship with matrices, with implications in linguistics, cryptography, and computer science. [ABSTRACT FROM AUTHOR]

In this paper, we prove the correspondence between finite dimensional representations of a simple Lie algebra and their associated characteristic polynomials. We will also define a monoid structure on these characteristic polynomials related to the tensor products of the representations. Furthermore, the factorization of characteristic polynomials sheds new light on the structure of simple Lie algebras and their Borel subalgebras. [ABSTRACT FROM AUTHOR]

We consider the price-optimal earliest arrival problem in public transit (POEAP) in which we aim to calculate the Pareto-set of journeys with respect to ticket price and arrival time in a public transportation network. Public transit fare structures are often a combination of various fare strategies such as, e.g., distance-based fares, zone-based fares or flat fares. The rules that determine the actual ticket price are often very complex. Accordingly, fare structures are notoriously difficult to model, as it is in general not sufficient to simply assign costs to arcs in a routing graph. Research into POEAP is scarce and usually either relies on heuristics or only considers restrictive fare models that are too limited to cover the full scope of most real-world applications. We therefore introduce conditional fare networks (CFNs), the first framework for representing a large number of real-world fare structures. We show that by relaxing label domination criteria, CFNs can be used as a building block in label-setting multi-objective shortest path algorithms. By the nature of their extensive modeling capabilities, optimizing over CFNs is NP-hard. However, we demonstrate that adapting the multi-criteria RAPTOR (McRAP) algorithm for CFNs yields an algorithm capable of solving POEAP to optimality in less than 400 ms on average on a real-world dataset. By restricting the size of the Pareto-set, running times are further reduced to below 10 ms.

Let p ∈ [1, ∞] and F : Set → Set be a functor with finite supports in the category Set of sets. Given a non-empty metric space (X, dX), we introduce the distance on the functor-space FX as the largest distance such that for every n ∈ ℕ and a ∈ Fn the map Xn → FX, f → Ff(a), is non-expanding with respect to the ℓp-metric on Xn. We prove that the distance is a pseudometric if and only if the functor F preserves singletons; is a metric if F preserves singletons and one of the following conditions holds: (1) the metric space (X, dX) is Lipschitz disconnected, (2) p = 1, (3) the functor F has finite degree, (4) F preserves supports. We prove that for any Lipschitz map f : (X, dX) → (Y, dY) between metric spaces the map is Lipschitz with Lipschitz constant Lip(Ff) ≤ Lip(f). If the functor F is finitary, has finite degree (and preserves supports), then F preserves uniformly continuous function, coarse functions, coarse equivalences, asymptotically Lipschitz functions, quasi-isometries (and continuous functions). For many dimension functions we prove the formula dim FpX ≤ deg(F) dim X. Using injective envelopes, we introduce a modification of the distance and prove that the functor Dist → Dist, , in the category Dist of distance spaces preserves Lipschitz maps and isometries between metric spaces. [ABSTRACT FROM AUTHOR]

Under suitable conditions on the unknown functions, solutions of Levi-Civita functional equations on commutative monoids with no prime ideals are exponential polynomials. This is not generally the case on commutative monoids with prime ideals. Here we describe the solutions of Levi-Civita equations on commutative monoids in which every prime ideal is tractable. Monoids with this property include those which are regular or generated by their squares, as well as many others. Our results also give the continuous solutions on topological commutative monoids with tractable prime ideals. [ABSTRACT FROM AUTHOR]

Human consciousness is characterized by constant transitions in time. On the other hand, what is consciously experienced always possesses the temporal feature of "now." In consciousness, "now" constantly holds different contents, yet it remains "now" no matter how far it goes. This duality is thematized in Husserlian phenomenology as "the standing-streaming now." Although this phrase appears contradictory in everyday language, it has a structure that can be clearly understood and formalized. In this paper, we show that this structure can be described as a monoid in category theory. Furthermore, monoids can be transformed into the coslice category, which corresponds to the way of perceiving present moments as juxtaposed in succession. The seemingly contradictory nature of the "now" as both flowing and standing can be precisely structured and comprehended through the monoid, while the perspective of the "now" as discrete points on a timeline can be effectively formalized using the coslice category. This framework helps us more precisely understand the differences between ordinary consciousness and meditative consciousness, specifically the experience of the "eternal now." We illustrate how the meditative states of consciousness presented in the early Buddhist scriptures (Pali Canon) and Dōgen's Shōbōgenzō remarkably reflect a monoid structure. [ABSTRACT FROM AUTHOR]

It has been recently observed that fundamental aspects of the classical theory of factorization can be greatly generalized by combining the languages of monoids and preorders. This has led to various theorems on the existence of certain factorizations, herein called ⪯-factorizations, for the ⪯-non-units of a (multiplicatively written) monoid H endowed with a preorder ⪯, where an element u ∈ H is a ⪯-unit if u ⪯ 1 H ⪯ u and a ⪯-non-unit otherwise. The "building blocks" of these factorizations are the ⪯-irreducibles of H (i.e., the ⪯-non-units a ∈ H that cannot be written as a product of two ⪯-non-units each of which is strictly ⪯-smaller than a); and it is interesting to look for sufficient conditions for the ⪯-factorizations of a ⪯-non-unit to be bounded in length or finite in number (if measured or counted in a suitable way). This is precisely the kind of questions addressed in the present work, whose main novelty is the study of the interaction between minimal ⪯-factorizations (i.e., a refinement of ⪯-factorizations used to counter the "blow-up phenomena" that are inherent to factorization in non-commutative or non-cancellative monoids) and some finiteness conditions describing the "local behavior" of the pair (H , ⪯). Besides a number of examples and remarks, the paper includes many arithmetic results, a part of which are new already in the basic case where ⪯ is the divisibility preorder on H (and hence in the setup of the classical theory). [ABSTRACT FROM AUTHOR]

We describe a practical algorithm for computing normal forms for semigroups and monoids with finite presentations satisfying so-called small overlap conditions. Small overlap conditions are natural conditions on the relations in a presentation, which were introduced by J. H. Remmers and subsequently studied extensively by M. Kambites. Presentations satisfying these conditions are ubiquitous; Kambites showed that a randomly chosen finite presentation satisfies the C (4) condition with probability tending to 1 as the sum of the lengths of relation words tends to infinity. Kambites also showed that several key problems for finitely presented semigroups and monoids are tractable in C (4) monoids: the word problem is solvable in O (min ⁡ { | u | , | v | }) time in the size of the input words u and v ; the uniform word problem for 〈 A | R 〉 is solvable in O (N 2 min ⁡ { | u | , | v | }) where N is the sum of the lengths of the words in R ; and a normal form for any given word u can be found in O (| u |) time. Although Kambites' algorithm for solving the word problem in C (4) monoids is highly practical, it appears that the coefficients in the linear time algorithm for computing normal forms are too large in practice. In this paper, we present an algorithm for computing normal forms in C (4) monoids that has time complexity O (| u | 2) for input word u , but where the coefficients are sufficiently small to allow for practical computation. Additionally, we show that the uniform word problem for small overlap monoids can be solved in O (N min ⁡ { | u | , | v | }) time. [ABSTRACT FROM AUTHOR]

In finance, optionality is a possible property of a financial contract giving the owner a choice between two or more assets. For example, a convertible bond has optionality because its owner must choose between having a bond or having some shares of stock. In mathematics, a binary operation acts on two elements in a set to produce a third element in that set. When a financial contract such as a convertible bond enjoys optionality between exactly two assets, then the arbitrage-free current value of the contract can potentially be treated as the outcome of a binary operation acting on the two current asset values. In this paper, we treat one of the two assets as riskless and demand that the binary operation linking the two current asset values always produces an arbitrage-free option price. In this context, we focus on the interplay between the properties of the risk-neutral density of the risky asset and the algebraic properties of the binary operation. [ABSTRACT FROM AUTHOR]

For a semigroup S, the covering number of S with respect to semigroups, σs (S), is the minimum number of proper subsemigroups of S whose union is S. This article investigates covering numbers of semigroups and analogously defined covering numbers of inverse semigroups and monoids. Our three main theorems give a complete description of the covering number of finite semigroups, finite inverse semigroups, and monoids (modulo groups and infinite semigroups). For a finite semigroup that is neither monogenic nor a group, its covering number is two. For all n ≥ 2, there exists an inverse semigroup with covering number n, similar to the case of loops. Finally, a monoid that is neither a group nor a semigroup with an identity adjoined has covering number two as well. [ABSTRACT FROM AUTHOR]

In this paper, we define measurable S -acts on monoids and construct a new category, namely Meas Act- S. We investigate the behavior of this category with respect to product, coproduct, pushout, pullback, equalizer and coequalizer. Next, we study the injectivity and complete injectivity in category Meas Act- S and show any measurable S -act can be embedded into complete injectivity. Moreover, we investigate the Skornjakov' theorem for measurable S-acts. [ABSTRACT FROM AUTHOR]

We define scalable monoids and prove their fundamental properties. Congruence relations on scalable monoids, direct and tensor products, subalgebras and homomorphic images of scalable monoids, and unit elements of scalable monoids are defined and investigated. A quantity space is defined as a commutative scalable monoid over a field, admitting a finite basis similar to a basis for a free abelian group. Observations relating to the theory of measurement of physical quantities accompany the results about scalable monoids. We conclude that the algebraic theory of scalable monoids and quantity spaces provides a rigorous foundation for quantity calculus. [ABSTRACT FROM AUTHOR]

Bergman (1974) found that for any prime number p, the endomorphism ring End(Zp x Zp²) is a semilocal ring which has p5 elements and can not be embedded in matrices over any commutative ring. Later on, Climent et al. (2011) found that each element of endomorphism ring End(Zp x Zp²) can be identified as a two by two matrix of Ep where the first and the second row entries belong to Zp and Zp² respectively. By this characterization, Long D. T., Thu D. T., and Thuc D. N. constructed a new RSA variant based on End(Zp x Zp²) (2013). In this paper, we state the characteristic of the endomorphism ring End (Zp² x Zp) and the RSA analogue cryptosystem based on it. [ABSTRACT FROM AUTHOR]

We study the functional equation for unknown functions f , g , h , k , ℓ : S → C , where S is a semigroup and m 1 , m 2 : S → C are multiplicative functions. The study is divided into two main parts: m 1 = m 2 and m 1 ≠ m 2 . In some cases we assume that one or more of the unknown functions is central and/or that S is a monoid. The solutions are found in all cases under the umbrella assumption that S is a commutative monoid. The continuous solutions on topological semigroups are also found. [ABSTRACT FROM AUTHOR]

We introduce two partially overlapping classes of pathwise dualities between interacting particle systems that are based on commutative monoids (semigroups with a neutral element) and semirings, respectively. For interacting particle systems whose local state space has two elements, this approach yields a unified treatment of the well-known additive and cancellative dualities. For local state spaces with three or more elements, we discover several new dualities. [ABSTRACT FROM AUTHOR]

We consider the Levi–Civita equation f(xy)=g1(x)h1(y)+g2(x)h2(y)f\left( {xy} \right) = {g_1}\left( x \right){h_1}\left( y \right) + {g_2}\left( x \right){h_2}\left( y \right) for unknown functions f, g1, g2, h1, h2 : S → ℂ, where S is a monoid. This functional equation contains as special cases many familiar functional equations, including the sine and cosine addition formulas. In a previous paper we solved this equation on groups and on monoids generated by their squares under the assumption that f is central. Here we solve the equation on monoids by two different methods. The first method is elementary and works on a general monoid, assuming only that the function f is central. The second way uses representation theory and assumes that the monoid is commutative. The solutions are found (in both cases) with the help of the recently obtained solution of the sine addition formula on semigroups. We also find the continuous solutions on topological monoids.

We completely classify all varieties of aperiodic monoids with central idempotents whose subvariety lattice is distributive. [ABSTRACT FROM AUTHOR]

Graph products of monoids provide a common framework for direct and free products, and graph monoids (also known as free partially commutative monoids). If the monoids in question are groups then any graph product is a group. For monoids that are not groups, regularity is perhaps the first and most important algebraic property that one considers: however, graph products of regular monoids are not in general regular. We show that a graph product of regular monoids satisfies the related, but weaker, condition of being abundant. More generally, we show that the classes of left abundant and left Fountain monoids are closed under graph product. As a very special case we obtain the earlier result of Fountain and Kambites that the graph product of right cancellative monoids is right cancellative. To achieve our aims we show that elements in (arbitrary) graph products have a unique Foata normal form, and give some useful reduction results; these may equally well be applied to groups as to the broader case of monoids. [ABSTRACT FROM AUTHOR]

Since topoi were introduced, there have been efforts putting mathematics into the context of topoi. Amongst known topoi, the topoi of sheaves or presheaves over a small category are of special interest. We have here as the base topos that of sheaves over a monoid M as a one object category. By means of closure operators we then obtain categories of sheaves related to the right ideals of M. These categories have already been studied but we give these categories a more thorough treatment and reveal some additional properties. Namely, for a weak topology determined by a right ideal I of M , we show that the category of sheaves associated to this topology is a subtopos of M S e t (the presheaves over M) and determine the Lawvere–Tierney topology yielding the same subtopos, which is the Lawvere–Tierney topology associated to the idempotent hull of the (not necessarily idempotent) closure operator associated to I. We will then find conditions under which the subcategory of separated objects turns out to be a topos, and in the last section, we find conditions under which the category of sheaves becomes a De Morgan topos. [ABSTRACT FROM AUTHOR]

Jones monoids n , for 1 < n , is a family of monoids relevant in knot theory. The purpose of this paper is to characterize the identities satisfied by the Jones monoid 5 . [ABSTRACT FROM AUTHOR]