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An example of a nonfinitely based involution monoid of order five has recently been discovered. We confirm that this example is, up to isomorphism, the unique smallest among all involution monoids. [ABSTRACT FROM AUTHOR]

In this paper, we show that all stalactic and taiga monoids of rank greater than or equal to 2 are finitely based and satisfy the same identities, that all sylvester monoids of rank greater than or equal to 2 are finitely based and satisfy the same identities and that all baxter monoids of rank greater than or equal to 2 are finitely based and satisfy the same identities. [ABSTRACT FROM AUTHOR]

Let (baxt n , ♯) be the Baxter monoid of finite rank n with Schützenberger's involution ♯ . In this paper, it is shown that (baxt n , ♯) admits a faithful representation by an involution monoid of upper triangular matrices over any semiring from a large class including the tropical semiring under the skew transposition. Then a transparent combinatorial characterization of the word identities satisfied by (baxt n , ♯) is given. Further, it is proved that (baxt n , ♯) is finitely based if and only if n ≠ 3 , and shown that the identity checking problem for (baxt n , ♯) is decidable in polynomial time. [ABSTRACT FROM AUTHOR]

Abstract Let ( hyp o n , ♯ ) be the hypoplactic monoid of finite rank n with Schützenberger’s involution ♯ . In this paper, we exhibit a faithful representation of ( hyp o n , ♯ ) as an involution monoid of upper triangular matrices over any semiring from a large class including the tropical semiring under the skew transposition. We then give a transparent combinatorial characterization of the word identities satisfied by ( hyp o n , ♯ ) . Further, we prove that ( hyp o n , ♯ ) is non-finitely based if and only if n = 2, 3 and give a polynomial time algorithm to check whether a given word identity holds in ( hyp o n , ♯ ) .Communicated by Scott Chapman [ABSTRACT FROM AUTHOR]

Let M n be the variety generated by all monoids of order n. It was known that M n is finitely based if n ⩽ 4 and non-finitely based if n ⩾ 6. However, the finite basis problem for M 5 remained open. In this paper, we show that the variety M 5 is finitely based by giving a finite identity basis. Therefore, M n is finitely based if and only if n ⩽ 5. [ABSTRACT FROM AUTHOR]

Let S be the commutative and idempotent semiring with additive identity 0 and multiplicative identity 1 . The tropical semiring T and the Boolean semiring B are common important examples of such semirings. Let U T n (S) be the semigroup of all n × n upper triangular matrices over S , both U T n ± (S) and U T n + (S) be subsemigroups of U T n (S) with 0 and/or 1 on the main diagonal, and 1 on the main diagonal respectively. It is known that U T 2 (T) is non-finitely based and U T 2 ± (S) is finitely based. Combining these results, the finite basis problems for U T n (T) and U T n ± (S) with n = 2 , 3 both as semigroups and involution semigroups under the skew transposition are solved. It is well known that the semigroups U T n + (S) and U T n + (B) are equationally equivalent. In this paper, we show that the involution semigroups U T n + (S) and U T n + (B) under the skew transposition are not equationally equivalent. Nevertheless, the finite basis problems for involution semigroups U T n + (S) and U T n + (B) share the same solution, that is, the involution semigroup U T n + (S) is finitely based if and only if n = 2 . [ABSTRACT FROM AUTHOR]

Textile‐based generators that can convert low‐grade energy from the human body or environment into sustainable electricity have generated immense scientific interest in self‐powered wearable applications. However, their low power density and environmental suitability have extremely restricted their portable applications in complex and mutable environments. Herein, an asymmetric sandwich structure between molybdenum disulfide (MoS2)‐carbonized silks (MCs) and MoS2/MXene–Cottons (MMCs) to construct efficient thermo–hydroelectric generators (THEGs) that synergistically harvest heat‐moisture energy to generate considerable electricity is rationally designed. Notably, the large surface area of MoS2/MXene van der Waals heterojunctions (vdWhs) enables efficient charge collection, and the vertical MoS2 nanosheet arrays supply abundant nanochannels for a highly efficient hydration effect, generating an output power density of 32.26 µW cm−2 after wetting with deionized water. Combined with the sensitive temperature recognition ability with a Seebeck coefficient of 23.5 µV K−1, the application possibilities of these prepared THEGs in the mutual conversion of fingertip temperature/language, and the monitoring of the human physiological state is foresee. [ABSTRACT FROM AUTHOR]