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In 1978, Doro, in his paper [S. Doro, Simple moufang loops, Math. Proc. Camb. Philos. Soc. 83 (1978) 377–392] published the following conjecture: If the nucleus of a Moufang loop is trivial, then the commutant is a normal subloop. By working in the multiplication group of the loop we prove that in case of finite Moufang loops with trivial nucleus, the commutant is normal if and only if it is trivial. [ABSTRACT FROM AUTHOR]

In [A.-P. Grecianu, A. Kvaschuk, A. G. Myasnikov and D. Serbin, Groups acting on hyperbolic Λ -metric spaces, Int. J. Algebra Comput. 25(6) (2015) 977–1042], the authors initiated a systematic study of hyperbolic Λ -metric spaces, where Λ is an ordered abelian group, and groups acting on such spaces. The present paper concentrates on the case Λ = ℤ n taken with the right lexicographic order and studies the structure of finitely generated groups acting on hyperbolic ℤ n -metric spaces. Under certain constraints, the structure of such groups is described in terms of a hierarchy (see [D. T. Wise, The Structure of Groups with a Quasiconvex Hierarchy : (AMS- 2 0 9) , Annals of Mathematics Studies (Princeton University Press, 2021)]) similar to the one established for ℤ n -free groups in [O. Kharlampovich, A. G. Myasnikov, V. N. Remeslennikov and D. Serbin, Groups with free regular length functions in ℤ n , Trans. Amer. Math. Soc. 364 (2012) 2847–2882]. [ABSTRACT FROM AUTHOR]

The aim of this paper is to characterize the formal power series which have purely periodic β-expansions in Pisot or Salem unit base under some condition. Furthermore, we will prove that if β is a quadratic Pisot unit base, then every rational f in the unit disk has a purely periodic β-expansion and discuss their periods. [ABSTRACT FROM AUTHOR]

Let us consider the group G = 〈x, y | xm = yn〉 with m and n nonzero integers. The set R(G) of representations of G over SL(2, ℂ) is a four-dimensional algebraic variety which is an invariant of G. In this paper the number of irreducible components of R(G) together with their dimensions are computed. We also study the set of metabelian representations of this family of groups. Finally, the behavior of the projection t : R(G) → X(G), where X(G) is the character variety of the group, and some combinatorial aspects of R(G) are investigated. [ABSTRACT FROM AUTHOR]

Differential modes provide examples of modes that do not embed as subreducts into semimodules over commutative semirings. The current paper studies differential modes, so-called Szendrei differential modes, which actually do embed into semimodules. These algebras form a variety. The main result states that the lattice of nontrivial subvarieties is dually isomorphic to the (nonmodular) lattice of congruences of the free commutative monoid on two generators. Consequently, all varieties of Szendrei differential modes are finitely based. [ABSTRACT FROM AUTHOR]

Let L be a free restricted Lie algebra and R a restricted ideal of L. Denote by u(L) the restricted enveloping algebra of L and by ω(L) the associative ideal of u(L) generated by L. The purpose of this paper is to identify the subalgebra R ∩ ωn(L)ω(R) in terms of R only. This problem is the analogue of the Fox problem for free groups. [ABSTRACT FROM AUTHOR]

Stallings showed that every finitely generated subgroup of a free group is canonically represented by a finite minimal immersion of a bouquet of circles. In terms of the theory of automata, this is a minimal finite inverse automaton. This allows for the deep algorithmic theory of finite automata and finite inverse monoids to be used to answer questions about finitely generated subgroups of free groups. In this paper, we attempt to apply the same methods to other classes of groups. A fundamental new problem is that the Stallings folding algorithm must be modified to allow for "sewing" on relations of non-free groups. We look at the class of groups that are amalgams of finite groups. It is known that these groups are locally quasiconvex and thus, all finitely generated subgroups are represented by finite automata. We present an algorithm to compute such a finite automaton and use it to solve various algorithmic problems. [ABSTRACT FROM AUTHOR]

A limit group is a finitely-generated subgroup of a fully residually free group. We prove in this paper the result announced in the title. [ABSTRACT FROM AUTHOR]

In this paper we construct a gathering process by the means of which we obtain new normal forms in braid groups. The new normal forms generalize Artin–Markoff normal forms and possess an extremely natural geometric description. In the two last sections of the paper we discuss the implementation of the introduced gathering process and the questions that arose in our work. This discussion leads us to some interesting observations, in particular, we offer a method of generating a random braid. [ABSTRACT FROM AUTHOR]

In this paper we prove that the pseudovariety of Abelian groups is hyperdecidable and moreover that it is completely tame. This is a consequence of the fact that a system of group equations on a free Abelian group with certain rational constraints is solvable if and only if it is solvable in every finite quotient. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the different notions of local topological density for subsets of the free monoid A*. We introduce the notion of weak completeness for a set X, relatively to an arbitrary submonoid M of A*. For the so-called strongly M-thin codes, we establish that weak completeness is equivalent to maximality in M. This constitutes a new generalization of a famous result due to Schützenberger. [ABSTRACT FROM AUTHOR]

In this paper we prove certain Hurwitz equivalence properties of B[sub n]. In particular we prove that for n = 3 every two Artin factorizations of [formula] of the form H[sub i[sub 1]] ⋯ H[sub i[sub 6]], F[sub j[sub 1]] ⋯ F[sub j[sub 6]] (with i[sub k], j[sub k] ∈ {1, 2}) where {H[sub 1], H[sub 2]}, {F[sub 1], F[sub 2]} are frames, are Hurwitz equivalent. The proof provided here is geometric, based on a newly defined frame type. The results will be applied to the classification of algebraic surfaces up to deformation. It is already known that there exist surfaces that are diffeomorphic to each other but are not deformations of each other (Manetti's example). We construct a new invariant based on Hurwitz equivalence of factorizations, to distinguish among diffeomorphic surfaces which are not deformation of each other. The main result of this paper will help us to compute the new invariant. [ABSTRACT FROM AUTHOR]

In this paper, a new computation system is defined by coupling a finite automaton with a deterministic Turing machine with one head and one tape that is infinite in one direction only. In a first part of the paper, it is shown that there is a Turing machine with five instructions for which it is possible to devise a finite automaton such that the resulting computation is able to simulate any Turing machine. In a second part of the paper it is shown that if the Turing machine has at most four instructions, whatever the finite automaton is, the halting of the resulting computation is always decidable. [ABSTRACT FROM AUTHOR]

In the present paper we generalize the concept of groups with triality and apply it to the theory of the Moufang, Bol and Bruck loops. Such generalizations allow us to reduce certain problems from the loop theory to problems in the theory of groups. [ABSTRACT FROM AUTHOR]

Thin Lie algebras are graded Lie algebras $L = \oplus_{i = 1}^{\infty}L_{i}$ with dim Li ≤ 2 for all i, and satisfying a more stringent but natural narrowness condition modeled on an analogous condition for pro-p-groups. The two-dimensional homogeneous components of L, which include L1, are named diamonds. Infinite-dimensional thin Lie algebras with various diamond patterns have been produced, over fields of positive characteristic, as loop algebras of suitable finite-dimensional simple Lie algebras, of classical or of Cartan type depending on the location of the second diamond. The goal of this paper is a description of the initial structure of a thin Lie algebra, up to the second diamond. Specifically, if Lk is the second diamond of L, then the quotient L/Lk is a graded Lie algebras of maximal class. In odd characteristic p, the quotient L/Lk is known to be metabelian, and hence uniquely determined up to isomorphism by its dimension k, which ranges in an explicitly known set of possible values: 3, 5, a power of p, or one less than twice a power of p. However, the quotient L/Lk need not be metabelian in characteristic two. We describe here all the possibilities for L/Lk up to isomorphism. In particular, we prove that k + 1 equals a power of two. [ABSTRACT FROM AUTHOR]

The main tools in the theory of n-ary hyperstructures are the fundamental relations. The fundamental relation on an n-ary hypersemigroup is defined as the smallest equivalence relation so that the quotient would be the n-ary semigroup. In this paper we study neutral elements in n-ary hypersemigroups and introduce a new strongly compatible equivalence relation on an n-ary hypersemigroup so that the quotient is a commutative n-ary semigroup. Also we determine some necessary and sufficient conditions so that this relation is transitive. Finally, we prove that this relation is transitive on an n-ary hypergroup with neutral (identity) element. [ABSTRACT FROM AUTHOR]

This paper studies the complexity of determining if a finite algebra generates a variety that satisfies various Maltsev conditions, such as congruence distributivity or modularity. For idempotent algebras we show that there are polynomial time algorithms to test for these conditions but that in general these problems are EXPTIME complete. In addition, we provide sharp bounds in terms of the size of two-generated free algebras on the number of terms needed to witness various Maltsev conditions, such as congruence distributivity. [ABSTRACT FROM AUTHOR]

In this paper we characterize when a Cayley automaton semigroup is finite, is free, is a left zero semigroup, is a right zero semigroup, is a group, or is trivial. We also introduce dual Cayley automaton semigroups and discuss when they are finite. [ABSTRACT FROM AUTHOR]

This paper introduces the foundations of the polynomial algebra and basic structures for algebraic geometry over the extended tropical semiring. Our development, which includes the tropical version for the fundamental theorem of algebra, leads to the reduced polynomial semiring — a structure that provides a basis for developing a tropical analogue to the classical theory of commutative algebra. The use of the new notion of tropical algebraic com-sets, built upon the complements of tropical algebraic sets, eventually yields the tropical algebraic Nullstellensatz. [ABSTRACT FROM AUTHOR]

Modes are idempotent and entropic algebras. Although it had been established many years ago that groupoid modes embed as subreducts of semimodules over commutative semirings, the general embeddability question remained open until Stronkowski and Stanovský's recent constructions of isolated examples of modes without such an embedding. The current paper now presents a broad class of modes that are not embeddable into semimodules, including structural investigations and an analysis of the lattice of varieties. [ABSTRACT FROM AUTHOR]

In this paper, we study the irreducible decompositions of the degeneracy loci of matrices which are defined by rank conditions on upper left submatrices. By introducing the concepts of standard and essential rank functions, we give an explicit classification of these degeneracy loci. Based on standard rank functions, we design an algorithm to write a degeneracy locus as a union of its irreducible components. This gives an answer to a problem raised by Sturmfels. [ABSTRACT FROM AUTHOR]

A semigroup is complex if it generates a variety with the property that every finite lattice is embeddable in its subvariety lattice. In this paper, subvariety lattices of varieties generated by small semigroups will be investigated. Specifically, all complex semigroups of minimal order will be identified. [ABSTRACT FROM AUTHOR]

Recently it has been noticed that many interesting combinatorial objects belong to a class of semigroups called left regular bands, and that random walks on these semigroups encode several well-known random walks. For example, the set of faces of a hyperplane arrangement is endowed with a left regular band structure. This paper studies the module structure of the semigroup algebra of an arbitrary left regular band, extending results for the semigroup algebra of the faces of a hyperplane arrangement. In particular, a description of the quiver of the semigroup algebra is given and the Cartan invariants are computed. These are used to compute the quiver of the face semigroup algebra of a hyperplane arrangement and to show that the semigroup algebra of the free left regular band is isomorphic to the path algebra of its quiver. [ABSTRACT FROM AUTHOR]

In this paper we produce a finite algebra which generates a variety with a PSPACE-complete membership problem. We produce another finite algebra with a γ function that grows exponentially. The results are obtained via a modification of a construction of the algebra A(T) that was introduced by McKenzie in 1996. [ABSTRACT FROM AUTHOR]

Evans defined quasigroups equationally, and proved a Normal Form Theorem solving the word problem for free extensions of partial Latin squares. In this paper, quasigroups are redefined as algebras with six basic operations related by triality, manifested as coupled right and left regular actions of the symmetric group on three symbols. Triality leads to considerable simplifications in the proof of Evans' Normal Form Theorem, and makes it directly applicable to each of the six major varieties of quasigroups defined by subgroups of the symmetric group. Normal form theorems for the corresponding varieties of idempotent quasigroups are obtained as immediate corollaries. [ABSTRACT FROM AUTHOR]

Let G be a finitely generated abelian group and G ≀ ℤ be the wreath product. In this paper, we classify all such groups G for which every automorphism of G ≀ ℤ has infinitely many twisted conjugacy classes. [ABSTRACT FROM AUTHOR]

There is a well-known correspondence between varieties of algebras and fully invariant congruences on the appropriate term algebra. A special class of varieties are those which are balanced, meaning they can be described by equations in which the same variables appear on each side. In this paper, we prove that the above correspondence, restricted to balanced varieties, leads to a correspondence between balanced varieties and inverse monoids. In the case of unary algebras, we recover the theorem of Meakin and Sapir that establishes a bijection between congruences on the free monoid with n generators and wide, positively self-conjugate inverse submonoids of the polycyclic monoid on n generators. In the case of varieties generated by linear equations, meaning those equations where each variable occurs exactly once on each side, we can replace the clause monoid above by the linear clause monoid. In the case of algebras with a single operation of arity n, we prove that the linear clause monoid is isomorphic to the inverse monoid of right ideal isomorphisms between the finitely generated essential right ideals of the free monoid on n letters, a monoid previously studied by Birget in the course of work on the Thompson group V and its analogues. We show that Dehornoy's geometry monoid of a balanced variety is a special kind of inverse submonoid of ours. Finally, we construct groups from the inverse monoids associated with a balanced variety and examine some conditions under which they still reflect the structure of the underlying variety. Both free groups and Thompson's groups Vn,1 arise in this way. [ABSTRACT FROM AUTHOR]

Hypersubstitutions are mappings which map operation symbols to terms. Terms can be visualized by trees. Hypersubstitutions can be extended to mappings defined on sets of trees. The nodes of the trees, describing terms, are labelled by operation symbols and by colors, i.e. certain positive integers. We are interested in mappings which map differently-colored operation symbols to different terms. In this paper we extend the theory of hypersubstitutions and solid varieties to multi-hypersubstitutions and colored solid varieties. We develop the interconnections between such colored terms and multi-hypersubstitutions and the equational theory of Universal Algebra. The collection of all varieties of a given type forms a complete lattice which is very complex and difficult to study; multi-hypersubstitutions and colored solid varieties offer a new method to study complete sublattices of this lattice. [ABSTRACT FROM AUTHOR]

Kaplansky's conjecture claims that the Jacobson radical $\mathcal{J}K[G]$ of a group algebra K[G], where K is a field of characteristic p > 0, coincides with its augmentation ideal $\mathcal{A}K[G]$ if and only if G is a locally finite p-group. By a theorem of Passman, if G is finitely generated and $\mathcal{J}K[G]=\mathcal{A}K[G]$ then any maximal subgroup of G is normal of index p. In the present paper, we consider one infinite series of finitely generated infinite p-groups (hence not locally finite p-groups), so called GGS-groups. We prove that their maximal subgroups are nonetheless normal of index p. Thus these groups remain among potential counterexamples to Kaplansky's conjecture. [ABSTRACT FROM AUTHOR]

Computably enumerable algebras are the ones whose positive atomic diagrams are computably enumerable. Computable algebras are the ones whose atomic diagrams are computable. In this paper we investigate computably enumerable algebras and provide several algebraic and computable theoretic distinctions of these algebras from the class of computable algebras. We give a characterization of computably enumerable but not computable algebras in terms of congruences and effective conjunctions of $\Pi_1^0$-sentences. Our characterization, for example, shows that computable conjunctions of negative atomic formulas true in a given c.e. algebra can be preserved in infinitely many of its homomorphic images. We also study questions on how expansions of algebras by finitely many new functions affect computable isomorphism types. In particular, we construct a c.e. algebra with unique computable isomorphism type but which has no finitely generated c.e. expansion. [ABSTRACT FROM AUTHOR]

One-relator presentations with torsion are characterized by relators which are proper powers of cyclically reduced words. Such groups have a spelling theorem which provides among other things a quick solution of the word problem. In this paper we consider other classes of one-relator presentations (which are necessarily torsion-free) with a similar spelling theorem. [ABSTRACT FROM AUTHOR]

We consider algebras over a field with generators x1,x2,…,xn subject to ${n\choose 2}$ square-free relations xixj=xkxl in which every product xpxq, p≠q, appears in one of the relations. The work of Gateva-Ivanova and Van den Bergh, motivated in particular by the study of set theoretic solutions of the Yang–Baxter equation, provided an important class of such algebras. In this special case the presentation satisfies the so-called cyclic condition that became an essential combinatorial tool in proving that these algebras share many strong ring theoretic properties of polynomial algebras in commuting variables. In this paper we describe the structure of algebras on four generators satisfying the cyclic condition. The emphasis is on some new unexpected features, not present in the motivating special classes. [ABSTRACT FROM AUTHOR]

An involuted semilattice -> is a semilattice with an involution -: S→S, i.e., -> satisfies $\bar{\bar a}=a$, and $\overline{a\vee b}={\bar a}\vee{\bar b}$. In this paper we study the properties of such semilattices. In particular, we characterize free involuted semilattices in terms of ordered pairs of subsets of a set. An involuted semilattice -,1> with greatest element 1 is said to be complemented if it satisfies a∨ā=1. We also characterize free complemented semilattices. We next show that complemented semilattices are related to ternary algebras. A ternary algebra -,0,ϕ,1> is a de Morgan algebra with a third constant ϕ satisfying $\phi={\bar \phi}$, and (a+ā)+ϕ=a+ā. If we define a third binary operation ∨ on T as a∨b=a*b+(a+b)*ϕ, then -,ϕ> is a complemented semilattice. [ABSTRACT FROM AUTHOR]

This paper considers the problem for what n and ω is the cyclically presented group Gn(ω) irreducible and trivial and studies the case when [Math Variable] and n≥5. [ABSTRACT FROM AUTHOR]

We consider two different types of bounded height condition for semigroups. The first one originates from the classical Shirshov's bounded height theorem for associative rings. The second which is weaker, in fact was introduced by Wolf and also used by Bass for calculating the growth of finitely generated (f.g.) nilpotent groups. Both conditions yield polynomial growth. We give the first two examples of f.g. semigroups which have bounded height and do not satisfy any nontrivial identity. One of these semigroups does not have bounded height in the sense of Shirshov and the other satisfies the classical bounded height condition. This develops further one of the main results of the author's paper (J. Algebra, 1993) where the first examples of f.g. semigroups of polynomial growth and without nontrivial identities were given. [ABSTRACT FROM AUTHOR]

A semiring with involution is a semiring equipped with an involutorial antiasutomorphism as a fundamental operation. The aim of the present paper is to determine the lattice of all varieties of idempotent and distributive semirings with involution. We start with the description of their structure, which is followed by a complete list of all subdirectly irreducibles. We make a heavy use of general results obtained recently by Dolinka and Vinčić [11] on involutorial Płonka sums. Applying these results and some further structural theorems, we construct the considered lattice. It turns out that it has exactly 64 elements. [ABSTRACT FROM AUTHOR]

Lyndon's group F[sup Z][x] is the free exponential group over the ring of integral polynomials Z[x]. This group, introduced by Lyndon in the 1960s, continues to be of interest to group theorists due to its importance in the study of first-order properties of free groups, in particular, equations over free groups. One of the crucial results of Lyndon's study was that the group F[sup Z][x] is fully residually F; i.e. for any finite collection of nontrivial elements in F[sup Z][x] there exists a homomorphism φ : F[sup Z][x] → F which is the identity on F and maps the given elements of F[sup Z][x] into nontrivial elements of F. The importance of F[sup Z][x] was further emphasized when Kharlampovich and Myasnikov proved in [3] that a finitely generated group is fully residually free if and only if it is embeddable into F[sup Z][x]. Lyndon's group and its subgroups play a vital role in the technique employed by O. Kharlampovich and A. Myasnikov in their solution of the famous Tarski problem on the decidability of the elementary theory of a free group (see [4, 5]). In this paper, we show that Lyndon's group is conjugately residually free, i.e. it is possible to map F[sup Z][x] to the free group F preserving the nonconjugacy of two elements. This result is a further step towards the understanding of the properties of F[sup Z][x]; moreover, it is closely related to the problem of "lifting solutions" of equations from F to F[sup Z][x], since our result implies that the solutions can indeed be "lifted" from F to F[sup Z][x] for equations of the type x[sup -1] c[sub 1] x = c[sub 2]. The structure of Lyndon's group, described by A. Myasnikov and V. Remeslennikov in [8], involves an infinite sequence of free constructions of a specific type, called free extensions of centralizers. For more results on residual properties of certain types of free constructions, see also the works of Ribes, Segal and Zalesskii (for example [9]). [ABSTRACT FROM AUTHOR]

The purpose of this work is to extend the classification of planar point configurations with low Waldschmidt constants initiated in [M. Dumnicki, T. Szemberg and H. Tutaj-Gasińska, Symbolic powers of planar point configurations II, J. Pure Appl. Algebra 220 (2016) 2001-2016] and continued in [M. Mosakhani and H. Haghighi, On the configurations of points in with the Waldschmidt constant equal to two, J. Pure Appl. Algebra 220 (2016) 3821-3825] for all values less than . As a consequence, we prove a conjecture of Dumnicki, Szemberg and Tutaj-Gasińska concerning initial sequences with low first differences. [ABSTRACT FROM AUTHOR]

In a paper concerning the relationship between coding theory and factorization theory Restivo, Salemi, and Sportelli made a conjecture that if subsets possess certain properties then they cannot form a factorization of a finite cyclic group. In this note it is shown that in fact such factorizations do exist. [ABSTRACT FROM AUTHOR]

We prove a constructive version of Tits' alternative for groups of quaternions with algebraic coefficients by bounding valuations of their entries considered as elements of a fraction field of an opportunely chosen Dedekind domain. [ABSTRACT FROM AUTHOR]

Much of the structure theory of inverse semigroups is based on constructing arbitrary inverse semigroups from groups and semilattices. It is known that E-unitary (or proper) inverse semigroups may be described as P-semigroups (McAlister), or inverse subsemigroups of semidirect products of a semilattice by a group (O'Carroll) or C[SUBu]-semigroups built over an inverse category acted upon by a group (Margolis and Pin). On the other hand, every inverse semigroup is known to have an E-unitary inverse cover (McAlister). The aim of this paper is to develop a similar theory for proper weakly left ample semigroups, a class with properties echoing those of inverse semigroups. We show how the structure of semigroups in this class is based on constructing semigroups from unipotent monoids and semilattices. The results corresponding to those of McAlister, O'Carroll and Margolis and Pin are obtained. [ABSTRACT FROM AUTHOR]

The Andrews--Curtis conjecture claims that every balanced presentation of the trivial group can be transformed into the trivial presentation by a finite sequence of "elementary transformations" which are Nielsen transformations together with an arbitrary conjugation of a relator. It is believed that the Andrews--Curtis conjecture is false; however, not so many possible counterexamples are known. It is not a trivial matter to verify whether the conjecture holds for a given balanced presentation or not. The purpose of this paper is to describe some nondeterministic methods, called Genetic Algorithms, designed to test the validity of the Andrews-Curtis conjecture. Using such algorithm we have been able to prove that all known (to us) balanced presentations of the trivial group where the total length of the relators is at most 12 satisfy the conjecture. In particular, the Andrews-Curtis conjecture holds for the presentation 〈x, y|xyx = yxy, x[SUP2] = y[SUP3]〉 which was one of the well known potential counterexamples. [ABSTRACT FROM AUTHOR]

In this paper we consider factorizing codes C over A, i.e. codes verifying the factorization conjecture by Schfützenberger. We give a description of the structure of the words in C ∩ a* (A\a)a*, a being a letter in A, by using a class of factorizations of the cyclic groups discovered by Hajós. [ABSTRACT FROM AUTHOR]

We prove that a metric space with subexponential asymptotic dimension growth has Yu's property A. [ABSTRACT FROM AUTHOR]

We improve Algorithm 5.1 of [Math. Comp. 86 (2017) 2519–2534] for computing all nonisomorphic skew left braces, and enumerate left braces and skew left braces of orders up to 868 with some exceptions. Using the enumerated data, we state some conjectures for further research. [ABSTRACT FROM AUTHOR]

Böröczky configurations of lines have been recently considered in connection with the problem of the containment between symbolic and ordinary powers of ideals. Here, we describe parameter families of Böröczky configurations of 1 3 , 1 4 , 1 6 , 1 8 and 2 4 lines and investigate rational points of these parameter spaces. [ABSTRACT FROM AUTHOR]

We determine the integers a, b ≥ 1 and the prime powers q for which the word map w(x, y) = xayb is surjective on the group PSL(2, q) (and SL(2, q)). We moreover show that this map is almost equidistributed for the family of groups PSL(2, q) (and SL(2, q)). Our proof is based on the investigation of the trace map of positive words. [ABSTRACT FROM AUTHOR]

An algebraically expandable class is a class of algebras axiomatizable by a set of sentences of the form ∀∃! ∧ p = q. We find all algebraically expandable classes in the variety of implication algebras. A representation result for finite implication algebras is proved, and we also give a characterization of the class of congruence permutable implication algebras. [ABSTRACT FROM AUTHOR]