Narrow Results

We give necessary and sufficient conditions for the efficiency of the direct product of finitely many finite monogenic monoids.

Let M be a monoid and let θ be an endomorphism of M. We prove that if the Bruck–Reilly extension BR(M, θ) is finitely presented, then the Bruck–Reilly extension BR(M, θ^m) is also finitely presented for all m ⩾ 1.

In this paper, the endomorphism monoid of circulant complete graph K(n,3) is explored explicitly. It is shown that AutK(n,3)) =D_n, the dihedral group of degree n. It is also shown that K(n,3) is unretractive when 3 does not divide n, End(K(3m,3)) =qEnd(K(3m,3)), sEnd(K(3m,3)) =Aut(K(3m,3)) and K(3m,3) is endomorphism-regular. The structure of End(K(3m,3)) is characterized and some enumerative problems concerning End(K(n,3)) are solved.

An endomorphism monoid of an algebra $\mathcal{A}$ is said to be a band if every endomorphism on $\mathcal{A}$ is an idempotent, and it is said to be a demi-band if every non-injective endomorphism on $\mathcal{A}$ is an idempotent. We precisely determine finite Kleene algebras whose endomorphism monoids are demi-bands and bands via Priestley duality.

Let A ⊆ B be an extension of integral domains and Γ a commutative, additive, cancellative, torsion-free monoid with Γ ∩ −Γ = {0}. Let B[Γ] be the semigroup ring of Γ over B and set Γ^∗ = Γ\{0}. Then R = A + B[Γ^∗] is a subring of B[Γ]. We investigate various factorization properties which are weaker than unique factorization in the domains of the form A + B[Γ^∗].

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.

A limit variety is a variety that is minimal with respect to being non-finitely based. Since the turn of the millennium, much attention has been given to the classification of limit varieties of aperiodic monoids. Seven explicit examples have so far been found, and the task of locating other examples has recently been reduced to two subproblems, one of which is concerned with monoids that satisfy the identity $xsxt\approx xsxtx$. We provide a complete solution to this subproblem by showing that there are precisely two limit varieties that satisfy this identity. One of them turns out to be the first example having infinitely many subvarieties. It is also deduced that the variety generated by any monoid of order 5 or less contains at most countably many subvarieties.