Slim groupoids

Slim groupoids are groupoids satisfying x(yz) ≈ xz. We find all simple slim groupoids and all minimal varieties of slim groupoids. Every slim groupoid can be embedded into a subdirectly irreducible slim groupoid. The variety of slim groupoids has the finite embeddability property, so that the word problem is solvable. We introduce the notion of a strongly nonfinitely based slim groupoid (such groupoids are inherently nonfinitely based) and find all strongly nonfinitely based slim groupoids with at most four elements; up to isomorphism, there are just two such groupoids. The work is a part of the research project MSM0021620839 financed by MSMT.     

Power semigroups of finite groups and the INFB property

We show that for any finite group, the involution semigroup consisting of all of its subsets is not inherently nonfinitely based.     

On the finite basis problem for the monoids of partial extensive injective transformations

Let \(PEI_n (POEI_n)\) be the monoid of all partial (order-preserving) extensive and injective transformations over a chain of order n. We give a sufficient condition under which a semigroup is nonfinitely based and apply this condition to show that the monoid \(PEI_3 (POEI_3)\) is nonfinitely based. This together with the results of Edmunds and Goldberg gives a complete answer to the finite basis problem for the monoid \(PEI_n (POEI_n)\): the monoid \(PEI_n (POEI_n)\) is nonfinitely based if and only if \(n\geqslant 3\). Furthermore, it is shown that the monoid \(PEI_n (POEI_n)\) is hereditarily finitely based if and only if \(n\leqslant 2\).     

The equational complexity of Lyndon’s algebra

The equational complexity of Lyndon’s nonfinitely based 7-element algebra lies between n − 4 and 2n + 1. This result is based on a new algebraic proof that Lyndon’s algebra is not finitely based. We prove that Lyndon’s algebra is inherently nonfinitely based relative to a rather rich class of algebras. We also show that the variety generated by Lyndon’s algebra contains subdirectly irreducible algebras of all cardinalities except 0, 1, and 4.     

Small inherently nonfinitely based finite semigroups

We show how to construct all ``forbidden divisors' for the pseudovariety of not inherently nonfinitely based finite semigroups. Several other results concerning finite semigroups that generate an inherently nonfinitely based variety that is miminal amongst those generated by finite semigroups are obtained along the way. For example, aside from the variety generated by the well known six element Brandt monoid \tb , a variety of this type is necessarily generated by a semigroup with at least 56 elements (all such semigroups with 56 elements are described by the main result). September 23, 1999     

On Freudenburg's counterexample to the Fourteenth Problem of Hilbert

In this paper we study Freudenburg's counterexample to the fourteenth problem of Hilbert and counterexamples derived from it. We shall construct a generating set of a nonfinitely generated G_a-invariant ring given in Freudenburg's counterexample by making use of an integral sequence which was constructed inductively by Freudenburg. This generating set shall be used in describing a generating set of a nonfinitely generated G_a-invariant ring given in Daigle and Freudenburg's counterexample. Using these generating sets, we shall determine the Hilbert series of the above Freudenburg's and Daigle and Freudenburg's nonfinitely generated G_a-invariant rings, and find that these Hilbert series are rational functions. Then we also show that the Hilbert series of nonfinitely generated invariant rings appearing in the author's linear counterexamples are rational functions.     

On identities of finite involution semigroups

We investigate the question of whether a finite involution semigroup is inherently nonfinitely based (INFB), which means that it is not contained in any finitely based locally finite variety. Although we fall short of a full characterization, we nevertheless clarify a number of interesting subcases.     

Locally Closed Semirings and Iteration Semirings

Locally closed semirings, iteration semirings and Conway semirings play an important role in the algebraic theory of semirings and theoretical computer science. Z. ésik and W. Kuich showed that a locally closed commutative semiring is an iteration semiring (is also a Conway semiring). By study of polynomial semirings and matrix semirings, we obtain new expressions of certain polynomials and show that all matrix semirings over a locally closed semiring are also locally closed, and so a locally closed semiring (which need not be commutative) is an iteration semiring.     

Locally Closed Semirings and Iteration Semirings

Locally closed semirings, iteration semirings and Conway semirings play an important role in the algebraic theory of semirings and theoretical computer science. Z. ésik and W. Kuich showed that a locally closed commutative semiring is an iteration semiring (is also a Conway semiring). By study of polynomial semirings and matrix semirings, we obtain new expressions of certain polynomials and show that all matrix semirings over a locally closed semiring are also locally closed, and so a locally closed semiring (which need not be commutative) is an iteration semiring.     

几类加法非正则半环

In this paper, we introduce Green＇s .-relations on semirings and define [left, right] adequate semirings to explore additively non-regular semirings. We characterize the semirings which are strong b-lattices of [left, right] skew-halfrings. Also, as further generalization, the semirings are described which are subdirect products of an additively commutative idempotent semiring and a [left, right] skew-halfring. We extend results of constructions of generalized Clifford semirings （given by M. K. Sen, S. K. MaRy, K. P. Shum, 2005） and the semirings which are subdirect products of a distributive lattice and a ring （given by S. Ghosh, 1999） to additively non-regular semirings.     

On Nonfinitely Based Varieties of Groups of Large Prime Exponent

For every sufficiently large odd p, we present a continual set of nonfinitely based varieties of groups of exponent p. The properties of these varieties make it possible to answer some open questions on varieties of groups.     

Identities of Semigroups of Triangular Matrices over Finite Fields

It is proved that the semigroup of all triangular n × n matrices over a finite field K is inherently nonfinitely based if and only if n > 3 and |K|> 2.     

链式条件下的半环的(∈,∈∨q(λ,μ))-模糊双理想

给出双诺特半环、双阿丁半环的概念。通过使用半环的双理想族建立半环的(∈,∈∨q(λ,μ))-模糊双理想。利用(∈,∈∨q(λ,μ))-模糊双理想刻画双诺特半环、双阿丁半环。     

FP-injective semirings,semigroup rings and Leavitt path algebras

In this paper we give characterisations of FP-injective semirings (previously termed “exact” semirings in work of the first author). We provide a basic connection between FP-injective semirings and P-injective semirings, and establish that FP-injectivity of semirings is a Morita invariant property. We show that the analogue of the Faith-Menal conjecture (relating FP-injectivity and self-injectivity for rings satisfying certain chain conditions) does not hold for semirings. We prove that the semigroup ring of a locally finite inverse monoid over an FP-injective ring is FP-injective and give a criterion for the Leavitt path algebra of a finite graph to be FP-injective.     

<Emphasis Type="Italic">p</Emphasis>-Ideals in <Emphasis Type="Italic">p</Emphasis>-Regular Semirings

In this paper, we introduce p-ideals in semirings. A new form of regularity, which is compatible with p-ideals, is defined. Our aim is to explore the possibilities of establishing an ideal theory in semirings, going alongside the existing literature of semiring theory.AMS Subject Classification (2000): Primary 16Y60     

Series parallel posets with nonfinitely generated clones

In 1986 Tardos proved that for the poset 1+2+2+2+1, the clone of monotone operations is nonfinitely generated. We generalize his result in the class of series parallel posets. We characterize the posets with nonfinitely generated clones in this class by the property that they have a retract of the form either 1+2+2+2+1, 2+2+1, or 1+2+2.Research partially supported by Hungarian National Foundation for Research under grant no. 1903.     

Nonfinitely Based Varieties of Right Alternative Metabelian Algebras

Since 1976, it is known from the paper by V. P. Belkin that the variety RA₂ of right alternative metabelian (solvable of index 2) algebras over an arbitrary field is not Spechtian (contains nonfinitely based subvarieties). In 2005, S. V. Pchelintsev proved that the variety generated by the Grassmann RA₂-algebra of finite rank r over a field ?, for char(?) ≠ 2, is Spechtian iff r = 1. We construct a nonfinitely based variety 𝔐 generated by the Grassmann 𝒱-algebra of rank 2 of certain finitely based subvariety 𝒱 ? RA₂ over a field ?, for char(?) ≠ 2, 3, such that 𝔐 can also be generated by the Grassmann envelope of a five-dimensional superalgebra with one-dimensional even part.     

Distributive lattice decompositions of semirings with a semilattice additive reduct

We describe the least distributive lattice congruence on the semirings in the variety of all semirings whose additive reduct is a semilattice, introduce the notion of a k-Archimedean semiring and characterize the semirings that are distributive lattices or chains of k-Archimedean semirings.