Weak implicative filters in quasi-ordered residuated systems

The concept of residuated relational systems ordered under a quasiorder relation was introduced in 2018 by S. Bonzio and I. Chajda as a structure A = 〈A, ·,→, 1, R〉, where (A, ·) is a commutative monoid with the identity 1 as the top element in this ordered monoid under a quasi-order R. The author introduced and analyzed the concepts of filters and implicative filters in this type of algebraic structures. In this article, the concept of weak implicative filters in a quasi-ordered residuated system is introduced as a continuation of previous researches. Also, some conditions for a filter of such system to be a weak implicative filter are listed.

Implicative filters in quasi-ordered residuated systems

The concept of residuated relational systems ordered under a quasiorder relation was introduced in 2018 by S. Bonzio and I. Chajda as a structure 𝒜 = 〈A, ·,→, 1, R〉, where (A, ·) is a commutative monoid with the identity 1 as the top element in this ordered monoid under a quasi-order R. The author introduced and analyzed the concepts of filters in this type of algebraic structures. In this article, as a continuation of previous author’s research, the author introduced and analyzed the concept of implicative filters in quasi-ordered residuated systems.

The action of Young subgroups on the partition complex

We study the restrictions, the strict fixed points, and the strict quotients of the partition complex $|\Pi_{n}|$ | Π n | , which is the $\Sigma_{n}$ Σ n -space attached to the poset of proper nontrivial partitions of the set $\{1,\ldots,n\}$ { 1 , … , n } .We express the space of fixed points $|\Pi_{n}|^{G}$ | Π n | G in terms of subgroup posets for general $G\subset \Sigma_{n}$ G ⊂ Σ n and prove a formula for the restriction of $|\Pi_{n}|$ | Π n | to Young subgroups $\Sigma_{n_{1}}\times \cdots\times \Sigma_{n_{k}}$ Σ n 1 × ⋯ × Σ n k . Both results follow by applying a general method, proven with discrete Morse theory, for producing equivariant branching rules on lattices with group actions.We uncover surprising links between strict Young quotients of $|\Pi_{n}|$ | Π n | , commutative monoid spaces, and the cotangent fibre in derived algebraic geometry. These connections allow us to construct a cofibre sequence relating various strict quotients $|\Pi_{n}|^{\diamond} \mathbin {\operatorname* {\wedge }_{\Sigma_{n}}^{}} (S^{\ell})^{\wedge n}$ | Π n | ⋄ ∧ Σ n ( S ℓ ) ∧ n and give a combinatorial proof of a splitting in derived algebraic geometry.Combining all our results, we decompose strict Young quotients of $|\Pi_{n}|$ | Π n | in terms of “atoms” $|\Pi_{d}|^{\diamond} \mathbin {\operatorname* {\wedge }_{\Sigma_{d}}^{}} (S^{\ell})^{\wedge d}$ | Π d | ⋄ ∧ Σ d ( S ℓ ) ∧ d for $\ell$ ℓ odd and compute their homology. We thereby also generalise Goerss’ computation of the algebraic André-Quillen homology of trivial square-zero extensions from $\mathbf {F}_{2}$ F 2 to $\mathbf {F}_{p}$ F p for $p$ p an odd prime.

Commutative semigroups whose endomorphisms are power functions

For any commutative semigroup S and positive integer m the power function $$f: S \rightarrow S$$ f : S → S defined by $$f(x) = x^m$$ f ( x ) = x m is an endomorphism of S. We partly solve the Lesokhin–Oman problem of characterizing the commutative semigroups whose all endomorphisms are power functions. Namely, we prove that every endomorphism of a commutative monoid S is a power function if and only if S is a finite cyclic group, and that every endomorphism of a commutative ACCP-semigroup S with an idempotent is a power function if and only if S is a finite cyclic semigroup. Furthermore, we prove that every endomorphism of a nontrivial commutative atomic monoid S with 0, preserving 0 and 1, is a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. We also prove that every endomorphism of a 2-generated commutative semigroup S without idempotents is a power function if and only if S is a subsemigroup of the infinite cyclic semigroup.

S-acts over a Well-ordered Monoid with Modular Congruence Lattice

This work relates to the structural act theory. The structural theory includes the description of acts over certain classes of monoids or having certain properties, for example, satisfying some requirement for the congruence lattice. The congruences of universal algebra is the same as the kernels of homomorphisms from this algebra into other algebras. Knowledge of all congruences implies the knowledge of all the homomorphic images of the algebra. A left $S$–act over monoid $S$ is a set $A$ upon which $S$ acts unitarily on the left. In this paper, we consider $S$–acts over linearly ordered and over well-ordered monoids, where a linearly ordered monoid $S$ is a linearly ordered set with a minimal element and with a binary operation $ \ max$, with respect to which $S$ is obviously a commutative monoid; a well-ordered monoid $S$ is a well-ordered set with a binary operation $ \ max$, with respect to which $S$ is also a commutative monoid. The paper is a continuation of the work of the author in co-authorship with M.S. Kazak, which describes $S$–acts over linearly ordered monoids with a linearly ordered congruence lattice and $S$-acts over a well-ordered monoid with distributive congruence lattice. In this article, we give the description of S-acts over a well-ordered monoid such that the corresponding congruence lattice is modular.