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Funding: The first author is supported by an Engineering and Physical Sciences Research Council grant (EP/P015638/1). The second author is supported by a Royal Society University Research Fellowship (URF\R\180015). Let H ≤ Sn be an intransitive group with orbits Ω1, Ω2, ... , Ωk. Then certainly H is a subdirect product of the direct product of its projections on each orbit, H|Ω1 x H|Ω2 x ... x H|Ωk. Here we provide a polynomial time algorithm for computing the finest partition P of the H-orbits such that H = Πc∈P H|c and we demonstrate its usefulness in some applications. Postprint

It is proved that any Azumaya semigroup algebra of an abundant semigroup over a field is a subdirect product of matrix algebras over Azumaya semigroup algebras of cancellative monoids. In particular, it is proved that for an abundant semigroup S with finitely many idempotents and a field K, $$K_0[S]$$ is Azumaya if and only if $$K_0[S]$$ is isomorphic to the direct sum of finite matrix algebras of Azumaya algebras of cancellative submonoids. This gives an affirmative answer to an open problem of Okninski in his monograph (Semigroup Algebra, Marcel Dekker, New York, 1991) for the case that the semigroup is a finite abundant semigroup.

Subdirectly irreducible universal algebras, i.e., algebras which are not decomposable into non-trivial subdirect product, play an important role in mathematics due to well-known Birkhoff Theorem which states that any algebra is a subdirect product of subdirectly irreducible algebras (in another terminology: every algebra is approximated by the subdirectly irreducible algebras). In view of this, it seems reasonable to study algebras with certain conditions on subdirectly irreducible algebras. One of the natural restrictions is the finiteness of all subdirectly irreducible algebras. More stronger restriction is boundness of orders of all subdirectly irreducible algebras. An act over a semigroup (it is also an automaton, and a unary algebra) is a set with an action of the given semigroup on it. The acts over a fixed semigroup form a variety whose signature coincides with self semigroup. On the other hand, it is a category whose morphisms are homomorphisms from one act into another. It is not difficult to see that the semigroups over which all subdirectly ireducible acts are finite, are exactly the semigroups over which all subdirectly irreducible acts are finitely approximated (in another terminology: residually finite). A more narrow class form semigroups over which all acts are approximated by acts of n or less elements where n is a fixed natural number. In 2000, I.B.Kozhukhov proved that all non-trivial acts over a semigroup S are approximated by two-element ones if and only if S is a semilattice (a commutative idempotent semigroup). In 2014, I.B.Kozhukhov and A.R.Haliullina proved that any semigroup with bounded orders of dubdirectly irreducible acts is uniformly locally finite, i.e., for every k, the orders of k-generated subsemigroups are bounded. In the work of I.B.Kozhukhov and A.V.Tsarev 2019, the authors described completely the abelian groups over which all acts are finitely approximated, and also abelian groups over which all acts are approximated by acts of bounded orders. In this work, we characterize the commutative semigroups over which all acts are approximated by acts consisted of n or less elements.

Let L_q(M) denote the lattice of all subquasivarieties of the quasivariety M under inclusion. There is a strong correlation between the properties of the lattice L_q(M) and algebraic systems from M. A. I. Maltsev first drew attention to this fact in a report at the International Congress of Mathematicians in 1966 in Moscow. In this paper, we obtain a characterization of the class of all distributive lattices, each of which is isomorphic to the lattice of some quasivariety of unars. A unar is an algebra with one unary operation. Obviously, any unar can be considered as an automaton with one input signal without output signals, or as an act over a cyclic semigroup. We construct partially ordered sets P∞ and P_s(s ∈ N0), where N0 is the set of all nonnegative integers. It is proved that a distributive lattice is isomorphic to the lattice L_q(M) for some quasivariety of unars M if and only if it is isomorphic to some principal ideal of one of the lattices 0(Ps)(s ∈ N0) or 0c(P∞), where 0(Ps)(s ∈ N0) is the ideal lattice of the poset Ps(s ∈ N0) and 0s(P∞) is the ideal lattice with a distinguished element 𝑐 of the poset P∞. The proof of the main theorem is based on the description of Q-critical unars. A finitely generated algebra is called Q-critical if it does not decompose into a subdirect product of its proper subalgebras. It was previously shown that each quasivariety of unars is determined by its Q-critical unars. This fact is often used to investigate quasivarieties of unars.

In the paper, we introduce the concept of a locally complete EMV-algebra and a locally $$\sigma $$ -complete EMV-algebra. They are generalizations of a complete and $$\sigma $$ -complete MV-algebra, respectively. We will find some representations for locally complete EMV-algebras with atomic idempotent elements. We prove that each EMV-algebra of this type is a subdirect product of directly indecomposable complete EMV-algebras with top element. Also, we find a necessary and sufficient condition for a locally complete EMV-algebra when its representing EMV-algebra with the greatest element 1 is complete. Finally, we investigate the Cantor–Bernstein-type theorems for locally $$\sigma $$ -complete EMV-algebras and $$\sigma $$ -complete EMV-algebras.

A class of rings is a formation whenever it contains all homomorphic images of its members and if it is subdirect product closed. In the present paper, it is shown that the lattice of all formations of finite rings is algebraic and modular. Let R be a finite commutative ring with an identity element. It is established that there is a one-to-one correspondence between the set of all invariant fuzzy prime ideals of R and the set of all fuzzy prime ideals of each ring of the formation generated by R.

A basic tool in universal algebra is that of a congruence. It has been shown that congruences can be defined for graphs with properties similar to their universal algebraic counterparts. In particular, a subdirect product of graphs and hence also a subdirectly irreducible graph, can be expressed in terms of graph congruences. Here the subdirectly irreducible graphs are determined explicitly. Using congruences, a graph theoretic version of the well-known Birkhoff Theorem from universal algebra is given. This shows that any non-trivial graph is a subdirect product of subdirectly irreducible graphs

In this paper, we introduce the notion of quasi-pseudo-BL algebras as the generalization of pseudo-BL algebras and quasi-pseudo-MV algebras. First, we investigate the properties of quasi-pseudo-BL algebras and show the subdirect product composition of any quasi-pseudo-BL algebra. Especially, some properties of good quasi-pseudo-BL algebras are presented. Second, we discuss the filters of quasi-pseudo-BL algebras and prove that there exists a bijective correspondence between normal filters and filter congruences on a quasi-pseudo-BL algebra. The properties of some special filters are also discussed. Finally, we study the ideals of quasi-pseudo-BL algebras and investigate some connections between ideals and filters of a quasi-pseudo-BL algebra.

In the paper weak isometries in directed groups are investigated. It is proved that for every weak isometryfin a directed groupGthe relationf(UL(x,y) ∩LU(x,y)) =UL(f(x),f(y)) ∩LU(f(x),f(y)) is valid for eachx,y∈G. The notions of an orthogonality of two elements and of a subgroup symmetry in directed groups are introduced and it is shown that each weak isometry in a 2-isolated directed group or in an abelian directed group is a composition of a subgroup symmetry and a right translation. It is also proved that stable weak isometries in a 2-isolated abelian directed groupGare directly related to subdirect decompositions of the subgroupG2= {2x;x∈G} ofG.

Extending Goursat's Lemma we investigate the structure of subdirect products of 3-factor direct products. We give several example constructions and then provide a structure theorem showing that every such group is essentially obtained by a combination of the constructions. The central observation in this structure theorem is that the dependencies among the group elements in the subdirect product that involve all three factors are of Abelian nature. In the spirit of Goursat's Lemma, for two special cases, we derive correspondence theorems between data obtained from the subgroup lattices of the three factors (as well as isomorphism between arising factor groups) and the subdirect products. Using our results we derive an explicit formula to count the number of subdirect products of the direct product of three symmetric groups., 14 pages

In this paper we endure the learning of Special Boolean-like rings(BLR) . In segment 1 we debate the assets of a Special Boolean-like ring. If R is a commutative ring with unity , we verify that R is a Special BLR allowing that R is a BLR. Further we display that a Special BLR is regular ⇔ it is a BR. A method is given to construct special Boolean rings from Boolean rings and certain modules over them. In section 2 we prove that a SBLR ‘ R’ is a subdirect product of a family of rings {Ri}, somewhere individually Ri is either a two component field or a four component BLR H4 or a zero-ring. In section 3 we discussed nearby the Jacobson radical J(R) of a Special BLR, R and demonstrate that J(R)=N( R) , where N(R) is the nilradical of R. As a moment of this, we illustration that every BR is semi simple. Finally we demonstrate that every special BLR which is semisimple, is a BR.

In this paper, we have introduced the notion of local and semilocal triangle algebras and propose the theorems that characterize these algebraic structures. Additionally, we have established the new properties of these algebraic structures and discussed the relations between local triangle algebras and some interval valued residuated lattice (IVRL)-filters, such as n-fold IVRL-extended integral filters and IVRL-extended maximal filters. The obtained results proved that the MTL-triangle algebra is a subdirect product of local triangle algebras. Moreover, a correlation was observed between the set of the dense elements and local triangle algebras. Finally, semilocal triangle algebras were introduced and assessed in detail, and an association was observed between the semilocal triangle algebras and quotient triangle algebras. submittedVersion

We consider necessary and sufficient conditions for finite generation and finite presentability for fiber products of free semigroups and free monoids. We give a necessary and sufficient condition on finite fiber quotients for a fiber product of two free monoids to be finitely generated, and show that all such fiber products are also finitely presented. By way of contrast, we show that fiber products of free semigroups over finite fiber quotients are never finitely generated. We then consider fiber products of free semigroups over infinite semigroups, and show that for such a fiber product to be finitely generated, the quotient must be infinite but finitely generated, idempotent-free, and $$\mathcal {J}$$ J -trivial. Finally, we construct automata accepting the indecomposable elements of the fiber product of two free monoids/semigroups over free monoid/semigroup fibers, and give a necessary and sufficient condition for such a product to be finitely generated.

The direct product $\mathbb{N}\times \mathbb{N}$ of two free monogenic semigroups contains uncountably many pairwise nonisomorphic subdirect products. Furthermore, the following hold for $\mathbb{N}\times S$, where $S$ is a finite semigroup. It contains only countably many pairwise nonisomorphic subsemigroups if and only if $S$ is a union of groups. And it contains only countably many pairwise nonisomorphic subdirect products if and only if every element of $S$ has a relative left or right identity element.

Loday and Ronco introduced the notion of a trioid and constructed the free trioid of rank 1. We construct the free commutative trioid of rank 1 and prove that the free commutative trioid of rank $$n>1$$ is a subdirect product of the free commutative semigroup of rank n and the free commutative trioid of rank 1. We also characterize the least commutative congruence, the least commutative dimonoid congruences, and the least commutative semigroup congruence on the free trioid.

In the present paper, the concept of a NQ-critical algebra is introduced. It is proved that if every connected component of a finitely generated algebra $$A$$ has the least subalgebra with respect to inclusion, then $$A$$ is either NQ-critical or a subdirect product of proper NQ-critical subalgebras.

Almost perfect commutative rings R are introduced (as an analogue of Bazzoni and Salce's almost perfect domains) for rings with divisors of zero: they are defined as orders in commutative perfect rings such that the factor rings R / R r are perfect rings (in the sense of Bass) for all non-zero-divisors r ∈ R . It is shown that an almost perfect ring is an extension of a T-nilpotent ideal by a subdirect product of a finite number of almost perfect domains. Noetherian almost perfect rings are exactly the one-dimensional Cohen–Macaulay rings. Several characterizations of almost perfect domains carry over practically without change to almost perfect rings. Examples of almost perfect rings with zero-divisors are abundant.

Let E / Q be an elliptic curve and let Q ( D 4 ∞ ) be the compositum of all extensions of Q whose Galois closure has Galois group isomorphic to a quotient of a subdirect product of a finite number of transitive subgroups of D 4 . In this article we first show that Q ( D 4 ∞ ) is in fact the compositum of all D 4 extensions of Q and then we prove that the torsion subgroup of E ( Q ( D 4 ∞ ) ) is finite and determine the 24 possibilities for its structure. We also give a complete classification of the elliptic curves that have each possible torsion structure in terms of their j-invariants.

The first two authors were supported by the National Science Foundation under Grant No. DMS 1500254. We prove the statement in the title and exhibit examples of quotients of arbitrary nilpotency class. This answers a question by Holt. Postprint

In this paper, we give a complete characterization of Leavitt path algebras which are graded Σ-V rings, that is, rings over which a direct sum of arbitrary copies of any graded simple module is graded injective. Specifically, we show that a Leavitt path algebra L over an arbitrary graph E is a graded Σ-V ring if and only if it is a subdirect product of matrix rings of arbitrary size but with finitely many non-zero entries over K or K [ x , x − 1 ] with appropriate matrix gradings. We also obtain a graphical characterization of such a graded Σ-V ring L. When the graph E is finite, we show that L is a graded Σ-V ring ⟺L is graded directly-finite ⟺L has bounded index of nilpotence ⟺ L is graded semi-simple. Examples show that the equivalence of these properties in the preceding statement no longer holds when the graph E is infinite. Following this, we also characterize Leavitt path algebras L which are non-graded Σ-V rings. Graded rings which are graded directly-finite are explored and it is shown that if a Leavitt path algebra L is a graded Σ-V ring, then L is always graded directly-finite. Examples show the subtle differences between graded and non-graded directly-finite rings. Leavitt path algebras which are graded directly-finite are shown to be directed unions of graded semisimple rings. Using this, we give an alternative proof of a theorem of Vas [33] on directly-finite Leavitt path algebras.

A semiring S is called a band orthodox semiring if $$E^+(S)$$ of all additive idempotents of S is a band semiring. Here we show that every band orthodox semiring is a subdirect product of a left band orthodox semiring and a right band orthodox semiring. Also we introduce the concept of quasi-spined product of a left normal band semiring and a d-inversive semiring with respect to a distributive lattice to study the structure of left normal band orthodox semirings.

This paper is a continuation of the work of Tom Head ‘Metatheorem for deriving fuzzy theorems from crisp versions’. The concept of natural extension is introduced which is then applied in the devel...

We show that a zero-symmetric near-ring $N$ is left regular if and only if $N $ is regular and isomorphic to a subdirect product of integral near-rings, where each component is either an Anshel-Clay near-ring or a trivial integral near-ring. We also show that a zero-symmetric near-ring is regular without nonzero nilpotent elements if and only if the multiplicative semigroup of N is a union of disjoint groups.

The algebraic notion of a “congruence” seems to be foreign to contemporary graph theory. We propound that it need not be so by developing a theory of congruences of graphs: a congruence on a graph ...

This paper is inspired by Mitsch, Petrich, Reilly’s work on E – inversive semigroups and inverse semigroups. We first introduce the notions of a good subdirect product and a good surjective subhomo...

A reduced Rickart ring is considered as a reduced ring R with an additional operation which associates to every element $$a \in R$$ the single idempotent e such that the ideal eR is the right annihilator of a. We discuss some elementary properties of this operation, prove that a ring is reduced and Rickart if and only if it is isomorphic to an associate ring in the sense of I. Sussman (a certain subdirect product of domains with “enough” idempotents), and present several equational axiom systems for reduced Rickart rings.

In this paper we study conjugacy separability of subdirect products of two free (or hyperbolic) groups. We establish necessary and sufficient criteria and apply them to fibre products to produce a finitely presented group G1 in which all finite index subgroups are conjugacy separable, but which has an index 2 overgroup that is not conjugacy separable. Conversely, we construct a finitely presented group G2 which has a non-conjugacy separable subgroup of index 2 such that every finite index normal overgroup of G2 is conjugacy separable. The normality of the overgroup is essential in the last example, as such a group G2 will always possess an index 3 overgroup that is not conjugacy separable. Finally, we characterize p-conjugacy separable subdirect products of two free groups, where p is a prime. We show that fibre products provide a natural correspondence between residually finite p-groups and p-conjugacy separable subdirect products of two non-abelian free groups. As a consequence, we deduce that the open question about the existence of an infinite finitely presented residually finite p-group is equivalent to the question about the existence of a finitely generated p-conjugacy separable full subdirect product of infinite index in the direct product of two free groups.

We investigate when fiber products of lattices are finitely generated and obtain a new characterization of bounded lattice homomorphisms onto lattices satisfying a property we call Dean’s condition (D) which arises from Dean’s solution to the word problem for finitely presented lattices. In particular, all finitely presented lattices and those satisfying Whitman’s condition satisfy (D). For lattice epimorphisms [Formula: see text], [Formula: see text], where [Formula: see text], [Formula: see text] are finitely generated and [Formula: see text] satisfies (D), we show the following: If [Formula: see text] and [Formula: see text] are bounded, then their fiber product (pullback) [Formula: see text] is finitely generated. While the converse is not true in general, it does hold when [Formula: see text] and [Formula: see text] are free. As a consequence, we obtain an (exponential time) algorithm to decide boundedness for finitely presented lattices and their finitely generated sublattices satisfying (D). This generalizes an unpublished result of Freese and Nation.

We give a definition of a generalized state operator (g-state operator) $$\sigma $$ on a residuated lattice X and a g-state residuated lattice $$(X,\sigma )$$ , by which the class of all g-state residuated lattices is proved to be a variety, and consider properties of g-state residuated lattices. We prove some fundamental results about them, such as characterizations of $$\sigma $$ -filters, extended $$\sigma $$ -filters, homomorphism theorems for g-state residuated lattices. Moreover, we show that every g-state residuated lattice is a subdirect product of subdirectly irreducible g-state residuated lattices.

An incline S is a commutative semiring where r + 1 = 1 for any r ∈ S . We note that the ideal lattice of an S-semimodule is naturally an S-semimodule and so is its congruence lattice when S is transitive. We prove that the categories of complete S-semimodules, together with dual functor, internal hom and tensor product, is a ⋆-autonomous category. We define the locally and globally maximal congruences which are related to Birkhoff subdirect product decomposition. We show that the categories of S-semimodules, algebraic S-semimodules and topological S-semimodules are equivalent. Finally, we get a sheaf representation of any S-semimodule.

We investigate a class of effect algebras that can be represented in the form $${\Gamma (H \overrightarrow{\times} G}$$ , (u, 0)), where $${H \overrightarrow{\times} G}$$ means the lexicographic product of an Abelian unital po-group (H, u) and an Abelian directed po-group G. We study conditions when an effect algebra is of this form. Fixing a unital po-group (H, u), the category of strongly (H, u)-perfect effect algebras is introduced and it is shown that it is categorically equivalent to the category of directed po-groups with interpolation. We prove some representation theorems of lexicographic effect algebras, including a subdirect product representation by antilattice lexicographic effect algebras.

For [Formula: see text] a positive integer and [Formula: see text] a finite set of finite algebras, let [Formula: see text] denote the largest [Formula: see text]-generated subdirect product whose subdirect factors are algebras in [Formula: see text]. When [Formula: see text] is the set of all [Formula: see text]-generated subdirectly irreducible algebras in a locally finite variety [Formula: see text], then [Formula: see text] is the free algebra [Formula: see text] on [Formula: see text] free generators for [Formula: see text]. For a finite algebra [Formula: see text] the algebra [Formula: see text] is the largest [Formula: see text]-generated subdirect power of [Formula: see text]. For every [Formula: see text] and finite [Formula: see text] we provide an upper bound on the cardinality of [Formula: see text]. This upper bound depends only on [Formula: see text] and these basic parameters: the cardinality of the automorphism group of [Formula: see text], the cardinalities of the subalgebras of [Formula: see text], and the cardinalities of the equivalence classes of certain equivalence relations arising from congruence relations of [Formula: see text]. Using this upper bound on [Formula: see text]-generated subdirect powers of [Formula: see text], as [Formula: see text] ranges over the [Formula: see text]-generated subdirectly irreducible algebras in [Formula: see text], we obtain an upper bound on [Formula: see text]. And if all the [Formula: see text]-generated subdirectly irreducible algebras in [Formula: see text] have congruence lattices that are chains, then we characterize in several ways those [Formula: see text] for which this upper bound is obtained.

In this paper a result of B. Leclerc and B. Monjardet concerning meet-projections in finite congruence-simple atomistic lattices is generalized. We prove that the result remains valid for any finite tolerance-simple lattice; moreover, we extend it to a type of subdirect product of such lattices, introducing the notion of a generalized oligarchy.

It is proved that all automorphism-invariant nonsingular right A-modules are injective if and only if the factor ring A/G(AA) of the ring A with respect to the right Goldie radical G(AA) is right strongly semiprime.

The article is devoted to the investigation of semirings with idempotent multiplication. General structure theorems for such semirings are proved. We focus on the study of the class \( \mathfrak{M} \) of all commutative multiplicatively idempotent semirings. We obtain necessary conditions under which semirings from \( \mathfrak{M} \) are subdirectly irreducible. We consider some properties of the variety \( \mathfrak{M} \). In particular, we show that \( \mathfrak{M} \) is generated by two of its subvarieties, defined by the identities 3x = x and 3x = 2x. We explore the variety \( \mathfrak{N} \) generated by two-element commutative multiplicatively idempotent semirings. It is proved that the lattice of all subvarieties of \( \mathfrak{N} \) is a 16-element Boolean lattice.

A theorem of single-sorted universal algebra asserts that every finite algebra can be represented as a product of a finite family of finite directly irreducible algebras. In this article, we show that the many-sorted counterpart of the above theorem is also true, but under the condition of requiring, in the definition of directly reducible many-sorted algebra, that the supports of the factors should be included in the support of the many-sorted algebra. Moreover, we show that the theorem of Birkhoff, according to which every single-sorted algebra is isomorphic to a subdirect product of subdirectly irreducible algebras, is also true in the field of many-sorted algebras.

We study when a pro-p subdirect product S ⩽ G1 × . . . × Gn is of type FPm for m ⩾ 2 for some special pro-p groups Gi. In particular we treat the case when Gi is a finitely generated non-trivial free pro-p product different from C2 ∐ C2 if p = 2 or a non-abelian pro-p group from the class $\mathcal{L}$ defined in [12].

We introduce the notion of a sturdy frame of abstract algebras which is a common generalization of a sturdy semilattice of semigroups, the sum of lattice ordered systems, the strong distributive lattice of semirings, the sturdy frame of type (2, 2) algebras and the strong b-lattice of semirings. Also, we give some properties and characterizations of the sturdy frame of abstract algebras. As an application, we study the sturdy distributive lattice of lattice ordered groups.

In this paper we investigate the Lie structure of the derived Lie superalgebra [K, K], with K the set of skew elements of a semiprime associative superalgebra A with superinvolution. We show that if U is a Lie ideal of [K, K], then either there exists an ideal J of A such that the Lie ideal [J \cap K,K] is nonzero and contained in U, or A is a subdirect sum of A', A'', where the image of U in A' is central, and A'' is a subdirect product of orders in simple superalgebras, each at most 16-dimensional over its center., arXiv admin note: substantial text overlap with arXiv:math/0701357

We consider unipotent, weakly cancellative, left cancellative, right cancellative, cancellative monoids and groups with the unary operation of mapping onto the identity element and bands with the identity mapping as unary operations. Then form Malcev products of each of the former with varieties of rectangular bands, semilattices, normal bands and all bands. Regarding the resulting semigroups with the induced unary operations, we characterize these classes, some of them we provide with a construction, and determine a basis for their implications, for they all turn out to be quasivarieties. We also determine their relationship which we clarify by an inclusion diagram.

For [Formula: see text] and for a ring [Formula: see text], the notation [Formula: see text] means that [Formula: see text] is nilpotent for all [Formula: see text], and [Formula: see text] means that [Formula: see text] has identity [Formula: see text] and [Formula: see text] is nil, where [Formula: see text] is the Jacobson radical of [Formula: see text]. In this paper, rings [Formula: see text] for which [Formula: see text] holds are completely characterized for integers [Formula: see text] up to [Formula: see text], for [Formula: see text] with [Formula: see text] odd, and for [Formula: see text], where [Formula: see text] and [Formula: see text] or [Formula: see text]. For an arbitrary integer [Formula: see text], we prove that [Formula: see text] for all rings [Formula: see text] if and only if [Formula: see text] is even with [Formula: see text], or [Formula: see text] is odd with [Formula: see text] and [Formula: see text].

In this paper, the authors study several classes of additively non-regular C-semirings whose additive idempotents are central, including the generalized C-rpp se- mirings, C-rpp semirings, generalized C-abundant semirings and C-abundant semirings. After introducing the concept of generalized C-rpp semirings, the authors obtain their equivalent characterizations, and show that a semiring is a generalized C-rpp semiring if and only if it is a strong b-lattice of additively left cancellative halfrings, and if and only if it is a subdirect product of a b-lattice and an additively left cancellative halfring. Also, the authors give the constructions of C-rpp semirings, generalized C-abundant se- mirings and C-abundant semirings. Consequently, the corresponding results of Clifiord semirings and generalized Clifiord semirings in (7) and (29) are extended and generalized.

The most natural seminearrings, i.e., the seminearrings of all self-maps of additive semigroups are necessarily multiplicatively regular but they need not be additively regular. The purpose of this paper is to investigate additively regular seminearrings. We mainly focus on the study of congruences in various types of additively regular seminearrings such as additively inverse, additively Clifford and Bandelt seminearrings. We deduce that for a restricted type of additively inverse seminearrings there exists an inclusion preserving bijective correspondence between the set of all normal congruences and that of all full k-ideals. Finally, we characterize those seminearrings which are the subdirect product of a distributive lattice and a zero symmetric near-ring.

We classify the pro-p subdirect products of type FPm of Demushkin or free pro-p groups, in both cases of non-trivial Euler characteristic.

Let N1 (N2) be the normal closure of a finite symmetrized set R1 (R2, respectively) in a finitely generated free group F = F(A). As is known, if Ri satisfies condition C(6), then the conjugacy problem is decidable in F/Ni. In the paper, it is proved that, if one adds to condition C(6) on the set R1 ∪ R2 the atoricity condition for the presentation 〈A | R1, R2〉, then the conjugacy problem is decidable in the group F/N1 ∩ N2 as well. In particular, for the decidability of the conjugacy problem in F/N1 ∩ N2, it is sufficient to assume that the set R1 ∪ R2 satisfies condition C(7).