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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 extend the concept of nil-extensions and retractive nil-extensions of different classes of semigroups to semirings and find a semiring analog of structure theorems on nil-extensions of different classes of additively regular semirings. [ABSTRACT FROM AUTHOR]

This paper is devoted to study some relationships of trusses with rings. We characterize abelian groups A such that every truss with retract A has exactly two non-commutative multiplications. We describe commutative rings with nontrivial multiplication having non-commutative extension by the ring of integers. [ABSTRACT FROM AUTHOR]

Ordered by set inclusion, the retracts of a lattice L together with the empty set form a bounded poset R e t (L) . By a grid we mean the direct product of two non-singleton finite chains. We prove that if G is a grid, then R e t (G) is a lattice. We determine the number of elements of R e t (G) . Some easy properties of retracts, retractions, and their kernels called retraction congruences of (mainly distributive) lattices are found. Also, we present several examples, including a 12-element modular lattice M such that R e t (M) is not a lattice. [ABSTRACT FROM AUTHOR]

We identify two key conditions that a subset A of a poset P may satisfy to guarantee the transfer of continuity properties from P to A. We then highlight practical cases where these key conditions are fulfilled. Along the way we are led to consider subsets of a given poset P whose way-below relation is the restriction of the way-below relation of P , which we call way-below preserving subposets. As an application, we show that every conditionally complete poset with the interpolation property contains a largest continuous way-below preserving subposet. Most of our results are expressed in the general setting of Z theory, where Z is a subset system. [ABSTRACT FROM AUTHOR]

Let H be an acylindrically hyperbolic group without nontrivial finite normal subgroups. We show that any finite system S of equations with constants from H is equivalent to a single equation. We also show that the algebraic set associated with S is, up to conjugacy, a projection of the algebraic set associated with a single splitted equation (such an equation has the form w.x1;: :: ; xn/D h, where w 2 F.X/, h 2 H). From this we deduce the following statement: Let G be an arbitrary overgroup of the above group H. Then H is verbally closed in G if and only if it is algebraically closed in G. These statements have interesting implications; here we give only two of them: If H is a noncyclic torsion-free hyperbolic group, then every (possibly infinite) system of equations with finitely many variables and with constants from H is equivalent to a single equation. We give a positive solution to Problem 5.2 from the paper [J. Group Theory 17 (2014), 29-40] of Myasnikov and Roman'kov: Verbally closed subgroups of torsion-free hyperbolic groups are retracts. Moreover, we describe solutions of the equation xnym D anbm in acylindrically hyperbolic groups (AH-groups), where a, b are non-commensurable jointly special loxodromic elements and n;m are integers with sufficiently large common divisor. We also prove the existence of special test words in AH-groups and give an application to endomorphisms of AH-groups. [ABSTRACT FROM AUTHOR]

A continuum is a nondegenerate, compact, connected, metric space. A topological property P is invariant under retraction provided that each retract of a continuum having P has P , and P is reversible under retraction by pseudo-deformation if the condition a subcontinuum of a continuum X has P implies that X has P. In this paper, we prove that the absence of R 3 -sets, the absence of R 4 -continua and the absence of s-points are reversible under retractions by pseudo-deformation, and the absence of R 3 -sets and the absence of R 4 -continua are invariant under retractions while the absence of s-points is not. [ABSTRACT FROM AUTHOR]

Let k be a field of characteristic zero and B a k-domain. Let R be a retract of B being the kernel of a locally nilpotent derivation of B. We show that if B = R ⊕ I for some principal ideal I (in particular, if B is a UFD), then B = R[1], i.e., B is a polynomial algebra over R in one variable. It is natural to ask that, if a retract R of a k-UFD B is the kernel of two commuting locally nilpotent derivations of B, then does it follow that B ≅ R[2]? We give a negative answer to this question. The interest in retracts comes from the fact that they are closely related to Zariski's cancellation problem and the Jacobian conjecture in affine algebraic geometry. [ABSTRACT FROM AUTHOR]

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We continue the work of [10], studying properties of digital images determined by fixed point invariants. We introduce pointed versions of invariants that were introduced in [10]. We introduce freezing sets and cold sets to show how the existence of a fixed point set for a continuous self-map restricts the map on the complement of the fixed point set.

FitzGerald identified four conditions (RI), (UR), (RI*) and (UR*) that are necessarily satisfied by an algebra if its monoid of endomorphisms has commuting idempotents. We show that these conditions are not sufficient, by giving an example of an algebra satisfying the four properties, such that its monoid of endomorphisms does not have commuting idempotents. This settles a problem presented by FitzGerald at the Conference and Workshop on General Algebra and Its Applications in 2013 and more recently at the workshop NCS 2018. After giving the counterexample, we show that the properties (UR), (RI*) and (UR*) depend only on the monoid of endomorphisms of the algebra, and that the counterexample we gave is in some sense the easiest possible. Finally, we list some categories in which FitzGerald's question has an affirmative answer. [ABSTRACT FROM AUTHOR]

In the study, algebraic structures and maps (category theory, morphisms and functors) that are inherently tied to the calculus of functors (orthogonal calculus) were explored. We emphasized on linear polynomial functors and generalized it to the n-polynomial functors as in the algebraic and topological settings where the topological settings talked about the Goodwillie and the Weiss case. [ABSTRACT FROM AUTHOR]

We describe retracts in free metabelian groups and obtain some structural results for verbally closed subgroups in free polynilpotent and free metabelian groups. [ABSTRACT FROM AUTHOR]

This paper is devoted to the investigation of cardinal invariants such as the hereditary density, hereditary weak density, and hereditary Lindelöf number. The relation between the spread and the extent of the space SP2(R,τ(A)) of permutation degree of the Hattori space is discussed. In particular, it is shown that the space SP2(R,τS) contains a closed discrete subset of cardinality c. Moreover, it is shown that the functor SPGn preserves the homotopy and the retraction of topological spaces. In addition, we prove that if the spaces X and Y are homotopically equivalent, then the spaces SPGnX and SPGnY are also homotopically equivalent. As a result, it has been proved that the functor SPGn is a covariant homotopy functor.

In this paper, we first consider the relationship between a polynomial ring B over a Noetherian domain R and the ring of invariants A of a G a -action on B , when A occurs as a retract of B. Next, we study retracts of a polynomial ring in general and address the questions of D.L. Costa raised in [5]. Finally, we examine the behaviour of ideals and certain properties of rings under retractions. [ABSTRACT FROM AUTHOR]

A set V in the tridisk D³ has the polynomial extension property if for every polynomial p there is a function ϕ on D³ so that ∥ϕ∥D³=∥p∥V and ϕ|V=p|V. We study sets V that are relatively polynomially convex and have the polynomial extension property. If V is one-dimensional, and is either algebraic, or has polynomially convex projections, we show that it is a retract. If V is two-dimensional, we show that either it is a retract, or, for any choice of the coordinate functions, it is the graph of a function of two variables. [ABSTRACT FROM AUTHOR]

We prove that for a compact X, the space Pf(X) is an absolute neighbourhood retract if and only if X is an absolute neighbourhood retract. Further, we demonstrate that a functor Pf preserves the property of a compact to be Q-manifold or Hilbert cube, preserves the property for leaves of maps to be absolute neighbourhood retracts in the class of compacts, to be Q-manifold or Hilbert cube (finite sum of Hilbert cubes). [ABSTRACT FROM AUTHOR]

Abstract In this paper, concepts of generalized algebraic information system (in short, GA-inf) and weak algebraic information system (in short, wA-inf) are introduced. Relationships among several algebraicities of information systems are explored. Main results are: (1) A dcpo D is an algebraic domain iff D is represented by an algebraic information system (in short, A-inf), and iff D is represented by a GA-inf; (2) every A-inf is a GA-inf and a wA-inf; (3) every C-inf S (D , B) induced by a domain with a basis B is a wA-inf; (4) the category WAINF of wA-infs and approximable mappings is equivalent to the category DOM of domains and Scott continuous functions; (5) every wA-inf S satisfying Condition (MT) is a retract of w S (the information structure induced by S) in category WAINF of wA-infs and approximable mappings. Particularly, C-inf S (D , B) induced by a domain D with a basis B is a retract of A-inf w S (D , B) in category WAINF. [ABSTRACT FROM AUTHOR]

We establish a Fourier inversion theorem for general connected, simply connected nilpotent Lie groups G=exp(g) by showing that operator fields defined on suitable sub-manifolds of g∗ are images of Schwartz functions under the Fourier transform. As an application of this result, we provide a complete characterisation of a large class of invariant prime closed two-sided ideals of L1(G) as kernels of sets of irreducible representations of G. [ABSTRACT FROM AUTHOR]

Background Interlaminar percutaneous endoscopic lumbar discectomy (PELD) is a minimally invasive technique to treat soft disc herniation. However, the currently used single working channel does not allow for safe nerve root retraction. This study aims to describe the rotate and retract technique for safe nerve root retraction during L5-S1 interlaminar PELD. Methods A total of 17 patients who underwent interlaminar PELD with the rotate and retract technique between November 2016 and August 2017 were retrospectively evaluated. Both pre- and postoperative visual analog scale (VAS) scores (back and leg) and Oswestry Disability Index (ODI) scores were used for clinical assessment. Results The mean preoperative VAS scores for back and leg pain were 3.84 ± 1.15 (range, 2–6) and 8.7 ± 0.2 (range, 8–10), respectively. The mean preoperative ODI score was 64.2 ± 13.2 (range, 48–90). The mean VAS scores for back and leg pain decreased to 1.24 ± 1.34 (range, 0–6) and 1.15 ± 0.65 (range, 0–4), respectively, at the last follow-up (P < 0.001). The mean ODI score also improved to 13.2 ± 5.1 (range, 9–29) at the last follow-up (P < 0.001). Conclusions The rotate and retract technique for interlaminar PELD is an effective maneuver for the treatment of L5-S1 disc herniations in selected patients if performed by experienced surgeons. Highlights • The rotate and retract technique for interlaminar PELD is an effective maneuver for treatment of L5-S1 disc herniations. • The surgeon should be sufficiently experienced to use this technique in properly selected cases. • The postoperative clinical results are satisfactory. [ABSTRACT FROM AUTHOR]

Fraïssé’s theorem characterizing the existence of universal homogeneous structures is a cornerstone of model theory. A categorical version of these results was developed by Droste and Göbel. Such an abstract version of Fraïssé theory allows to construct unusual objects that are far away from the usual structures. In this paper we are going to derive sufficient conditions for a comma category to contain universal homogeneous objects. Using this criterion, we characterize homogeneous structures that possess universal homogeneous endomorphisms. The existence of such endomorphisms helps to reduce questions about the full endomorphism monoid to the self-embedding monoid of the structure. As another application we characterize the retracts of homogeneous structures that are induced by universal homogeneous retractions. This extends previous results by Bonato, Delić, Mudrinski, Dolinka, and Kubiś. [ABSTRACT FROM AUTHOR]

For a set of sorts S and an S-sorted signature Σ we prove that a profinite Σ-algebra, i.e. a projective limit of a projective system of finite Σ-algebras, is a retract of an ultraproduct of finite Σ-algebras if the family consisting of the finite Σ-algebras underlying the projective system is with constant support. In addition, we provide a categorial rendering of the above result. Specifically, after obtaining a category where the objects are the pairs formed by a nonempty upward directed preordered set and by an ultrafilter containing the filter of the final sections of it, we show that there exists a functor from the just mentioned category whose object mapping assigns to an object a natural transformation which is a retraction. [ABSTRACT FROM AUTHOR]

The monoid Cb of name substitutions originated by Pitts in name abstraction, and the notion of a finitely supported Cb-set appeared in the study of models of homotopy type theory in the works of Gabbay and Pitts. On the other hand, retracts and retractions play a crucial role in most branches of mathematics as well as in computer science where partial morphisms need to be completed. Retracts are the subobjects whose related inclusion morphism have a left inverse, called retraction. In this paper, we study the retracts and retractions of cyclic finitely supported Cb-sets. We find the general definition of retractions from a cyclic Cb-set, and give necessary conditions under which retractions exist. Also, fix-simple retracts of a cyclic Cb-set are characterized. Further, the cyclic finitely supported Cb-sets all whose subobjects are retract, are studied. In particular, we give a necessary condition for a cyclic finitely supported Cb-set to be retractable. [ABSTRACT FROM AUTHOR]

In this paper we investigate Tychonoff spaces which are retracts of paratopological groups. A strong necessary condition for that is the existence of a certain binary operation on the space (called a τ -twister), which was introduced in [2,3] . Some general theorems are established which imply that βω is not a retract of a paratopological group. We also notice, using some deep results of V.V. Uspenskij, that the space ω 1 of countable ordinals is not a retract of any topological group (see Fact 3 ). [ABSTRACT FROM AUTHOR]

We study algebraically and verbally closed subgroups and retracts of finitely generated nilpotent groups. A special attention is paid to free nilpotent groups and the groups UT(Z) of unitriangular ( n× n)-matrices over the ring Z of integers for arbitrary n. We observe that the sets of retracts of finitely generated nilpotent groups coincides with the sets of their algebraically closed subgroups. We give an example showing that a verbally closed subgroup in a finitely generated nilpotent group may fail to be a retract (in the case under consideration, equivalently, fail to be an algebraically closed subgroup). Another example shows that the intersection of retracts (algebraically closed subgroups) in a free nilpotent group may fail to be a retract (an algebraically closed subgroup) in this group. We establish necessary conditions fulfilled on retracts of arbitrary finitely generated nilpotent groups. We obtain sufficient conditions for the property of being a retract in a finitely generated nilpotent group. An algorithm is presented determining the property of being a retract for a subgroup in free nilpotent group of finite rank (a solution of a problem of Myasnikov). We also obtain a general result on existentially closed subgroups in finitely generated torsion-free nilpotent with cyclic center, which in particular implies that for each n the group UT(Z) has no proper existentially closed subgroups. [ABSTRACT FROM AUTHOR]

We give a characterization of countable discrete subspace A of a topological space X such that there exists a (linear) continuous mapping φ : C p ⁎ ( A ) → C p ( X ) with φ ( y ) | A = y for every y ∈ C p ⁎ ( A ) . Using this characterization we answer two questions of A. Arhangel'skii. Moreover, we introduce the notion of well-covered subset of a topological space and prove that for well-covered functionally closed subset A of a topological space X there exists a linear continuous mapping φ : C p ( A ) → C p ( X ) with φ ( y ) | A = y for every y ∈ C p ( A ) . [ABSTRACT FROM AUTHOR]

Let $A_{2}$ be a free associative algebra or polynomial algebra of rank two over a field of characteristic zero. The main results of this paper are the classification of noninjective endomorphisms of $A_{2}$ and an algorithm to determine whether a given noninjective endomorphism of $A_{2}$ has a nontrivial fixed element for a polynomial algebra. The algorithm for a free associative algebra of rank two is valid whenever an element is given and the subalgebra generated by this element contains the image of the given noninjective endomorphism. [ABSTRACT FROM PUBLISHER]